UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


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THE   CORNELL   MATHEMATICAL    SERIES 

LUCIEN   AUGUSTUS   WAIT  •  •  •  General  Editor 

(bBNIOR  PRUFE68UK  OF  MATHEMATICS  IN  CORNELL  UNIVERSITY) 


The  Cornell  Mathematical  Series, 
lucien  augustus  wait, 

{Senior  Profeuor  of  Mathematics  in  Cornell  Umvenity,) 
GENERAL  EDITOR. 


This  series  is  designed  primarily  to  meet  the  needs  of  students  in  En- 
gineering and  Architecture  in  Cornell  University  ;  and  accordingly  many 
practical  problems  in  illustration  of  the  fundamental  principles  play  an  early 
and  important  part  in  each  book. 

While  it  has  been  the  aim  to  present  each  subject  in  a  simple  manner, 
yet  rigor' of  treatment  has  been  regarded  as  more  important  than  simplicity, 
and  thus  it  is  hoped  that  the  series  will  be  acceptable  also  to  general  students 
of  Mathematics. 

The  general  plan  and  many  of  the  details  of  each  book  were  discussed  at 
meetings  of  the  mathematical  staff.  A  mimeographed  edition  of  each  vol- 
ume was  used  for  a  term  as  the  text-book  in  all  classes,  and  the  suggestions 
thus  brought  out  were  fully  considered  before  the  work  was  sent  to  press. 

The  series  includes  the  following  works  : 
ANALYTIC  GEOMETRY.     By  J.  H,  Tanner  and  Joseph  Allen. 

DIFFERENTIAL  CALCULUS.    By  James 'McMahon  and  Viroil  Sntder. 

INTEGRAL  CALCULUS.    By  D.  A.  Mhbrat. 


ELEMENTS 


DIFFEREI^TIAL    CALCULUS 


JAMES   McMAHON,  A.M.  (dublin) 

ASSISTANT   PROFESSOR  OF  MATHEMATICS  IN  CORNELL  UNIVERSITY 
AND 

VIRGIL   SNYDER,  Ph.D.  (gottingen) 

INSTRUCTOR   IN    MATHEMATICS   IN   CORNELL    UNIVERSITY 


UNIV.  OF  California 

ATLOS  ANG  ttrs  UBRARY 

NEW   YORK  :•  CINCINNATI  •:•  CHICAGO 

AMERICAN     BOOK    COMPANY 


Copyright,  1898,  by 
JAMES   McMAHON  and  VIKGIL  SNYDER. 


ELE.    DIP.    OAU 
W.    P.   3 


Libfary 

PREFACE 


This  book  is  primarily  designed  as  a  text-book  for  the 
classes  in  the  Calculus  at  Cornell  University  and  other  insti- 
tutions in  which  the  object  and  extent    of   the  work  are 

'>  similar.  For  the  engineering  students  at  Cornell,  Differen- 
tial Calculus  is  taught  during  the  winter  term  of  the  fresh- 
man year ;  the  students  are  then  familiar  with  Analytic 
Geometry,  and  many  properties  of  the  conies  can  be  sup- 
posed known. 

When   use   is   made   of   Cartesian   coordinates,   they  are 

V.     always  assumed  to  be  rectangular. 

K  As  an  apology  for  adding  still  another  work  to  a  field  in 
which  the  literature  is  already  extensive,  it  may  be  said 
that  probably  no  other  book  has  just  the  scope  ,of  this  one. 
Many  of  the  works  are  too  brief,  and  omit  rigorous  proofs  as 
being  too  difficult  for  the  average  student,  while  the  more 
extensive  treatises  have  too  much  for  a  student  to  master  in 
the  allotted  time. 

While  the  chapter  on  fundamental  principles  is  a  long 
one,  nothing  more  is  introduced  than  is  necessary  for  sub- 
sequent parts  of  the  work  ;  and  it  is  hoped  that  the  matter 
is  so  arranged  that  the  student  will  not  find  it  difficult 
reading. 

In  the  chapter  on  expansion  of  functions  unusual  stress  is 
laid  upon  convergence  and  the  calculation  of  the  remainder ; 
and  numerous  examples  are  discussed  to  illustrate  the  prin- 
ciples. 

T 


^ 


vi  PREFACE 

The  chapter  on  asymptotes  is  perhaps  unusually  long,  as 
the  subject  is  so  presented  that  the  form  of  any  infinite 
branch  can  be  readily  determined  from  its  approximate 
equation,  and  the  process  is  fully  illustrated  both  in  this 
chapter  and  in  a  later  one  on  curve  tracing. 

Quite  a  full  discussion  is  given  to  the  form  of  a  curve  in 
the  vicinity  of  a  singular  point,  the  method  of  expansion 
being  extensively  used. 

No  list  of  "  higher  plane  curves  "  has  been  prepared,  since 
the  subject,  as  usually  given,  is  properly  a  part  of  Analytic 
Geometry.  A  chapter  on  that  subject  is  contained  in  the 
Analytic  Geometry  of  this  series.  The  occasional  marginal 
references  [A.  G.]  are  to  this  book. 

No  specific  acknowledgments  to  other  works  have  been 
given  ;  for  although  various  works  have  been  consulted,  the 
main  inspiration  has  come  from  the  class  room  and  from 
extensive  consultation  with  our  colleagues. 

Many  of  the  examples  have  been  selected  from  other 
books,  but  a  large  number  are  new.  When  original  exam- 
ples have  been  taken  from  recent  works,  acknowledgment 
of  the  source  is  made. 

We  acknowledge  our  indebtedness  to  the  other  authors 
of  this  series  for  their  hearty  cooperation ;  to  our  colleagues, 
Dr.  J.  I.  Hutchinson  and  Dr.  G.  A.  Miller,  for  the  keen 
interest  they  have  taken  in  the  work  and  for  their  assistance 
in  verifying  examples  and  reading  proof  ;  to  Mr.  Peter 
Field,  Fellow  in  Mathematics,  for  solving  the  entire  list  of 
exercises ;  and  to  Mr.  V.  T.  Wilson,  Instructor  in  Drawing 
in  Sibley  College,  for  drawing  the  figures.  Every  figure  in 
the  book  is  new,  and  drawn  to  scale,  except  that  in  some 
cases  vertical  ordinates  are  proportionately  foreshortened 
to  fit  the  page. 


CONTENTS 


INTRODUCTION 

ABTIOUC  PAGE 

1.  Number 1 

2.  Operations 1 

3.  Expressions 3 

4.  Functions 4 

5.  Constants  and  variables 7 

6.  Continuous  variable ;  continuous  function 7 

CHAPTER  I 
Fundamental  Principles 

7.  Limit  of  a  variable y 

8.  Infinitesimals  and  infinites 10 

9.  Fundamental  theorems  concerning  infinitesimals,  and  limits  in 

general 11 

10.  Comparison  of  variables 17 

11.  Comparison  of  infinitesimals,  and  of  infinites.     Orders  of  mag- 

nitude   18 

12.  Order  of  magnitude  of  expressions  involving  infinitesimals  or 

infinites 20 

13.  Useful  illustrations  of  infinitesimals  of  different  orders       .        .  25 

14.  Continuity  of  functions 28 

15.  Comparison  of  simultaneous  infinitesimal  increments  of  two 

related  variables 33 

16.  Definition  of  a  derivative 37 

17.  Geometric  illustrations  of  a  derivative 38 

18.  The  operation  of  differentiation 41 

19.  Increasing  and  decreasing  functions 43 

20.  Algebraic  test  of  the  intervals  of  increasing  and  decreasing        .  45 

21.  Differentiation  of  a  function  of  a  function 46 

22.  Differentiation  of  inverse  functions 47 

vii 


viii  CONTENTS 

CHAPTER  n 

Differentiation  of  the  Elementary  Forms 

ARTICLK  PAOB 

23.  Recapitulation ...    50 

24.  Differentiation  of  the  product  of  a  constant  and  a  variable  .     50 

25.  Differentiation  of  a  sum 51 

26.  Differentiation  of  a  product 52 

27.  Differentiation  of  a  quotient 53 

28.  Differentiation  of  a  commensurable  power  of  a  function     .        .     54 

29.  Elementary  transcendental  functions 58 

30.  Differentiation  of  loga  x  and  log,,  u 59 

31.  Differentiation  of  the  simple  exponential  function       .        .        .61 

32.  Differentiation  of  the  general  exponential  function     .        .        .61 

33.  Differentiation  of  an  incommensurable  power      .        .        .        .62 

34.  Differentiation  of  sin  m 63 

35.  Differentiation  of  cos  M .        .        .63 

36.  Differentiation  of  tan  u 64 

37.  Differentiation  of  cot  u  .        .        .        .      ' 64 

38.  Differentiation  of  sec  u 65 

39.  Differentiation  of  esc  u 65 

40.  Differentiation  of  vers  u 65 

Differentiation  of  the  Inverse  Trigonometric  Functions 

41.  Differentiation  of  sin-^u 66 

42.  Differentiation  of  cos-^  u 67 

43.  Differentiation  of  tan-^u 68 

44.  Differentiation  of  cot~^u 68 

45.  Differentiation  of  sec'^u 69 

46.  Differentiation  of  csc-^u 69 

47.  Differentiation  of  vers"^  m 70 

List  of  Elementary  Forms 71 


CHAPTER  m 
Successive  Differentiation 

48.  Definition  of  nth  derivative 73 

49.  Expression  for  the  nth  derivative  in  certain  cases        .        .        .74 

50.  I^ibnitz's  theorem  concerning  the  nth  derivative  of  a  product    .     75 

51.  Successive  i-derivatives  of  y  when  neither  variable  is  independent    77 


CONTENTS  IX 


CHAPTER   IV 
Expansion  of  Functions 

A8TICLK  PAOV 

52.  Introductory 81 

53.  Convergence  and  divergence  of  series 82 

54.  General  test  for  interval  of  convergence 83 

55.  Interval  of  equivalence.     Remainder  after  n  terms      .        .        .86 

56.  Maclaurin's  expansion  of  a  function  in  power-series    .        .        .87 

57.  Development  of  /(x)  in  powers  of  a:  —  a       •         .         .        .        .93 

58.  Remainder 95 

59.  RoUe's  theorem Q    95~ 

60.  Form  of  remainder  in  development  of /(x)  in  powers  of  x—a. 

Lagrange's  form  .        , 95 

61.  Another  expression  for  the  remainder.     Cauchy's        .        .        .97 

62.  Form  of  remainder  in  Maclaurin's  series 99 

63.  Lemma.    —  =  0,    n  =  oo 100 

n! 

64.  Remainder  in  the  development  of  a*,  sin  z,  cos  a; .        .        .        .  100 

65.  Taylor's  series 101 

Proof  of  the  binomial  theorem 102 

Calculation  of  natural  logarithms 105 

66.  Theorem  of  mean  value.    Increment  of  function  in  terms  of 

increment  of  variable 107 

Increment  of  the  increment 108 

lim     A^y^rffy j^g 

Ax  =  0  Ax'^     (/x2 

67.  Illustration :  to  find  the  development  of  a  function  when  that  of 

its  derivative  is  known 109 

Series  for  tan-'x,  and  calculation  of  7r 110 

Series  for  sin"^x,  and  calculation  of  ir 112 


CHAPTER  V 
Indeterminate  Forms 

68.  Definition  of  an  indeterminate  form 115 

69.  Indeterminate  forms  may  have  determinate  values      .        .        .116 

70.  Evaluation  by  transformation 118 

71.  Evaluation  by  development 119 

72.  Evaluation  by  differentiation 121 

73.  Evaluation  of  the  indeterminate  form  —■...».  125 


X  CONTENTS 

ARTICLE  FAGB 

74.  Evaluation  of  the  form  oo  •  0 126 

75.  Evaluation  of  the  form  oo  —  oo 126 

76.  Evaluation  of  the  form  1" 130 

77.  Evaluation  of  the  forms  0»,  oo" 130 

CHAPTER  VI 
Mode  of  Variation  of  Functions  of  One  Variable 

78.  Review  of  increasing  and  decreasing  functions  .        .        .132 

79.  Turning  values  of  a  function 132 

30.     Critical  values  of  the  variable 134 

81.  Method  of  determining  whether  <^'  (x)  changes  sign         .        .  134 

82.  Second  method  of  determining  whether  <^'(x)  changes  sign 

in  passing  through  zero .         .  137 

83.  Conditions  for  maxima  and  minima  derived  from  Taylor's 

theorem 139 

84.  Application  to  rational  polynomials 141 

85.  Maxima  and  minima  occur  alternately 143 

86.  Simplifications  that  do  not  alter  critical  values  .        .        .  143 

87.  Geometric  problems  in  maxima  and  minima     ....  144 

CHAPTER  VII 
Rates  and  Differentials 

88.  Rates.     Time  as  independent  variable 151 

89.  Abbreviated  notation  for  rates 154 

90.  Differentials  often  substituted  for  rates 156 

CHAPTER  Vm 
Differentiation  of  Functions  of  More  than  One  Variable 

91.  Definition  of  continuity    .        .- 158 

92.  Rate  of  variation.    Partial  derivatives 159 

93.  Geometric  illustration 160 

94.  Simultaneous  variation  of  x  and  y;  total  rate  of  variation  of  z  161 

95.  Language  of  differentials 164 

96.  One  variable  a  function  of  the  other 165 

97.  Differentiation  of  implicit  functions ;  relative  variation  that 
z  constant 166 


CONTENTS  XI 

« 

ABTIOLB  PAGE 

98.  Functions  of  more  than  two  variables 168 

99.  One  or  two  relations  between  the  three  variables  x,  y,z.        .     169 

100.  Euler's  theorem ;  relation  between  a  homogeneous  function 

and  its  partial  derivatives 171 

CHAPTER  IX 
Successive  Paktial  Differentiation 

101.  Successive  differentiation  of  functions  of  two  variables .         .  173 

102.  Order  of  differentiation  indifferent 175 

103.  Extension  of  Taylor's  theorem  to  expansion  of  two  variables  .  178 

104.  Significance  of  remainder 180 

105.  Form  corresponding  to  Maclaurin's  theorem  ....  180 

CHAPTER  X 
Maxima  and  Minima  of  Functions  of  Two  Variables 

106.  Definition  of  maximum  and  minimum  of  functions  of  two 

variables 183 

107.  Determination  of  maxima  and  minima 183 

108.  Conditional  maxima  and  minima 191 

Implicit  functions     .........  193 

CHAPTER  XI 
Change  of  the  Variable 


109.  Interchange  of  dependent  and  independent  variables 

110.  Change  of  the  dependent  variable    . 

111.  Change  of  the  independent  variable 

112.  Change  of  two  independent  variables 

113.  Change  of  three  independent  variables     . 

114.  Application  to  higher  derivatives     . 


198 

199 
199 
201 
204 
205 


Xii  CONTENTS 

APPLICATIONS   TO   GEOMETRY 

CHAPTER  XII 
Tangents  and  Normals 

ARTICLE  PAOX 

115.  Geometric  meaning  of  -^ 208 

dx 

116.  Equation  of  tangent  and  normal  at  a  given  point    .        .        .     208 

117.  Length  of  tangent,  normal,  subtangent,  subnormal         .         .     209 

Polar  Coordinates 

118.  Meaning  of  p—       .        . 213 

119.  Relation  between  ^  and  p  — 213 

fix  dp 

120.  Length  of  tangent,  normal,  polar  subtangent,  and  polar  sub- 

normal    213 

CHAPTER  XIII 

Derivative  of  an  Arc,  Area,  Volume  and  Surface 
of  Revolution 


121.  Derivative  of  an  arc 

122.  Trigonometric  meaning  of  — ,   — 

d^     dy 

123.  Derivative  of  the  volume  of  a  solid  of  revolution 

124.  Derivative  of  a  surface  of  revolution 

125.  Dei'ivative  of  arc  in  polar  coordinates 

126.  Derivative  of  area  in  polar  coordinates    . 


216 
217 

218 
218 
219 
220 


CHAPTER  XIV 
Asymptotes 

127.  Hyperbolic  and  parabolic  branches  ......    221 

128.  Definition  of  a  rectilinear  asymptote 221 

Determination  of  Asymptotes 

129.  Method  of  limiting  intercepts 221 

130.  Method  of  inspection.     Infinite  ordinates,  asymptotes  parallel 

to  axes 224 

131.  Infinite  ordinates  are  asymptotes 226 

132.  Method  of  substitution.     Oblique  asymptotes  .        .        .    227 


CONTENTS  Xlll 


133.  Number  of  asymptotes      .         .    "     .        .        . 

134.  Method  of  expansion.     Explicit  functions 

135.  Method  of  expansion.     Implicit  f  mictions 

136.  Curvilinear  asymptotes 

137.  Examples  of  asymptotes  of  transcendental  curves 

138.  Asymptotes  in  polar  coordinates 

139.  Determination  of  asymptotes  to  polar  curves  . 


CHAPTER  XV 
Direction  of  Bending.    Points  of  Inflexion 


PAGE 

229 
230 
234 
236 
237 
239 
240 


140.  Concavity  upward  and  downward 243 

141.  Algebraic  test  for  positive  and  negative  bending     .         .         .  244 

142.  Analytical  proof  of  the  test  for  the  direction  of  bending         .  247 

143.  Concavity  and  convexity  towards  the  axis       ....  248 

144.  Concavity  and  convexity  ;  polar  coordinates    ....  249 

CHAPTER  XVI 
Contact  and  Curvature 

145.  Order  of  contact 252 

146.  Number  of  conditions  implied  by  contact        ....  253 

147.  Contact  of  odd  and  of  even  order 254 

148.  Circle  of  curvature .  255 

149.  Length  of  radius   of   curvature;    coordinates   of  renter   of 

curvature 255 

150.  Second  method 257 

151.  Direction  of  radius  of  curvature 259 

152.  Other  forms  iox  R 260 

1.53.     Total  curvature  of  a  given  arc;  average  curvature  .        .        .  261 

154.  Measure  of  cui-vature  at  a  given  point 261 

155.  Curvature  of  an  arc  of  a  circle 262 

156.  Curvature  of  osculating  circle .  263 

157.  Direct  derivation  of  the  expressions  for  k  and  R  in  polar 

coordinates 265 

EvoLUTES  and  Involutes 

158.  Definition  of  an  evolute 267 

159.  Properties  of  the  evolute 269 


XIV  CONTENTS 

CHAPTER   XVU 
Singular  Points 

ARTICLE  FAOK 

160.     Definition  of  a  singular  point 275 

1(51.     Determination  of  singular  points  of  algebraic  curves       .         .  275 

162.  Multiple  points 277 

163.  Cusps 278 

164.  Conjugate  points 281 

CHAPTER  XVni 
Curve  Tracing 

165.  Statement  of  problem 283 

166.  Trace  of  curve  x*  -  jj*  +  d  xy^  =  0 284 

167.  Form  of  a  curve  near  the  origin        ......  288 

Another  proof 292 

169.  Oblique  branch  through  origin.     Expansion  of  y  in  ascend- 

ing powers  of  a; 292 

170.  Branches  touching  either  axis 295 

171.  Two  branches  oblique ;  a  third  touching  z-axis        .         .        .  297 

172.  Approximation  to  form  of  infinite  branches     ....  299 

173.  Curve  tracing;  polar  coordinates 303 

CHAPTER   XIX 
Envelopes 

174.  Family  of  curves 307 

175.  Envelope  of  a  family  of  curves 308 

176.  Envelope  touches  every  curve  of  the  family     ....  309 

177.  Envelope  of  normals  of  a  given  curve 310 

178.  Two  parameters,  one  equation  of  condition     ....  311 

Appendix 314 

Answers .  327 

Index 335 


INTEODUCTION 


In  this  general  introductory  chapter  some  terms  of  fre- 
quent use  in  subsequent  work  will  be  briefly  recalled  to  mind 
and  illustrated,  and  their  meaning  somewhat  extended. 

1.  Number.  Mathematics  is  concerned  with  the  study  of 
numbers  and  their  relations  to  each  other.  Numbers  are 
represented  by  letters,  as  a,  5,  c,  a;,  y,  a,  etc.,  or  by  figures, 
1,  2,  7,  VIl,  -8,  f,  etc. 

2.  Operations.     The  process  of  obtaining  a  number  from 
\  other  numbers  by  any  definite  rule  is  called  an  operation. 

Addition,  subtraction,  multiplication,  division,  raising  to 
integral  powers  and  extracting  roots  are  called  algebraic 
operations.  All  other  operations  are  transcendental;  thus 
taking  the  cosine  or  the  logarithm  of  a  given  number  is  a 
transcendental  operation. 

An  inverse  operation  is  the  undoing  of  what  was  done  by 
the  corresponding  direct  operation,  and  it  ends  where  the 
direct  operjition  began.  It  may  also  be  defined  as  an  opera- 
tion the  effect  of  which  the  direct  operation  simply  annuls. 

E.g.,  if  y  =  ax  +  b,  then  y  is  produced  by  performing  a  certain  opera- 
tion Upon  X,  viz.  multiplying  it  by  a  and  adding  h  to  the  product.  The 
invei-se  operation  consists  in  expressing  x  in  terms  of  y,  and  this  is  done 
by  subtracting  h  from  y  and  dividing  the  difference  by  a. 

1 


2  DIFFERENTIAL   CALCULUS  [Int. 

Again,  \i  y  =  sin  x,  tlien  x  =  sin~i^,  which  is  variously  read,  "  x  is  the 
angle  whose  sine  is  y,"  "  x  is  the  inverse  sine  of  y,"  "  x  is  the  anti-sine 
of  y."  The  relation  between  the  direct  and  inverse  operators  may  be 
shown  by  the  identity 

sin  (sin-^y)  =  y, 

which  expresses  the  truism  that  y  is  the  sine  of  any  of  the  angles  whose 
sine  is  y. 

Similarly,  2  =  logg  9,  which  is  read  the  logarithm  of  9  to  the  base  3, 
hence  9  =  log3~i2,    read  the  anti-logarithm  of  2  to  the  base  3; 

in  which  as  before  the  operating  symbol  logg  is  transferred  from  one 
member  to  the  other  by  annexing  the  index  of  inversion  (  —  1),  in  con- 
ventional analogy  with  the  familiar  transference  of  a  multiplier  in  such 
equations  as  y  =  mx,  x  =  m~'^y.  Here  in~^  has  a  meaning  in  itself;  but 
loga"^  has  no  meaning  unless  followed  by  a  number.  Thus  (log3  2)~i  and 
logg"^  2  have  distinct  meanings ;  the  former  inverts  the  number  logg  2, 
while  the  latter  inverts  the  operator  logg. 

Two  operations  are  said  to  be  successive  when  one  is  per- 
formed upon  the  result  of  the  other  ;  e.g.,  log  sin  a;  means  : 
take  the  sine  of  x  and  then  the  logarithm  of  sin  x. 

Successive  operations  are  called  commutative  when  their 
sequence  may  be  altered  without  changing  the  result. 

Thus  any  successive  operations  of  multiplication  and  division  are  com- 
mutative ;  e.g.,  a'h'C-—  — a.--c-h\  but  taking  the  sine  and  the  logar 

d  d 

rithm  are  not  commutative,  for  sin  log  x  ^  log  sin  x. 

Another  property  of  certain  operations  may  be  first  illus- 
trated numerically : 

3(12  +  6)  =  3.12  +  3.6, 
122x62  =  (12x6)2. 

In  the  first  illustration,  the  result  of  multiplying  12  +  6 
by  3  is  the  Same  whether  the  number  12  and  6  be  first 
added,  and  then  their  sum  be  multiplied  by  3,  or  12  and  6  be 
multiplied  separately  by  3,  and  the  results  added.  In  the 
second  illustration,  the  operation  of  squaring  may  be  per- 


2-3.]  INTRODUCTION  3 

formed  upon  the  separate  factors,  and  the  results  multiplied, 
or  it  may  be  performed  after  the  multiplication. 

These  facts  are  expressed  by  saying  that  multiplication  is 
distributive  as  to  addition,  and  that  involution  is  distributive 
as  to  multiplication. 

The  general  definition  of  distributive  operations  may  now 
be  stated  as  follows  : 

If  one  operation  consists  in  combining  several  elements 
into  one  result,  a  second  operation  is  distributive  as  to  the 
first  when  the  final  result  is  the  same,  whether  the  second 
operation  be  performed  upon  the  result  of  the  first  opera- 
tion, or  upon  the  several  elements  of  the  first  operation,  and 
then  these  results  combined  by  the  first  operation. 

Since  (12x«)  +  3  is  not  equal  to  (12  +  3)(6  +  3),  and  log (216  +  36) 
is  not  equal  to  log  216  +  log  36,  hence  addition  is  not  distributive 
as  to  multiplication,  nor  taking  the  logarithm,  as  to  addition. 

3.  Expressions-  Any  combination  of  letters  and  symbols 
used  to  denote  a  number  is  an  expression.  It  may  be  called 
an  expression  or  a  number,  according  as  the  thought  is  of 
the  symbol,  or  of  the  value  which  the  symbol  represents. 

An  expression  is  algebraic  when  there  are  implied  no  other 
operations  upon  the  numbers  of  which  it  is  composed  than 
algebraic  operations,  and  when  none  of  these  operations  is 
repeated  an  infinite  number  of  times.  An  infinite  series 
involves,  in  general,  only  the  operations  of  addition  and 
multiplication,  but  it  is  not  called  an  algebraic  expression. 


E.g.,  3  J,  4  ?/  +  17  x_i/z,  a*,  //*  +  x-,  Va^x^y*  are  algebraic  expressions. 

Expressions  which  imply  other  operations  than  a  finite 
number  of  algebraic  ones  are  called  transcendental  expres- 
sions. 

H.g.,  sin  z,  7",  log  (2  a:  +  y),  are  transcendental  expressions. 

niFF.  CAI.C.  — 2 


4  DIFFERENTIAL   CALCULUS  [Int. 

An  algebraic  expression  is  7'ational,  wlien  it  does  not  con- 
tain radicals  ;  surd,  or  irrational  when  it  contains  radicals 
or  fractional  exponents  ;  entire,  or  integral,  when  it  has  only 
a  numerical  denominator.  An  expression  may  be  integral 
as  to  some  of  its  letters,  and  fractional  as  to  others. 

7    rv*    A     ft  y  /        I        /i       , 

E.g.,     J—I-—  is  integral  as  to  x,  y,  b,  but  fractional  as  to  a  and 

a  —  4c 
to  c. 

An  expression  is  symmetric  as  to  any  of  its  letters,  when 
its  value  remains  the  same,  however  these  letters  be  inter- 
changed. 

E.g.,  xyz,  x  -\-  y  +  z  are  symmetric  as  to  x,  y,  z,  or  to  any  two  of  them ; 
w  +  x  —  y  —  zis  symmetric  as  to  w  and  x,  and  as  to  y  and  z,  but  not  as  to 
X  and  y,  w  and  y,  to  and  z,  nor  x  and  z.  " 

An  expression  is  said  to  be  transformed  when  it  is  changed 
ill  form  but  not  in  value  ;  it  is  developed  or  expanded  when 
transformed  into  a  series. 

4.  Functions.  If  a  number  is  so  related  to  other  numbers 
that  its  value  depends  upon  their  values,  it  is  a  function  of 
those  numbers  ;  the  function  is  explicit  when  directly  ex- 
pressed in  terms  of  those  numbers  ;  implicit,  when  not  so 
expressed. 

Thus  y  is  an  explicit  function  of  x  when  the  equation  of 
relation  between  x  and  y  is  solved  for  y. 

The  numbers  upon  which  the  value  of  the  function  depends 
are  called  the  arguments  of  the  function. 

E.g.,  in  u  =  ^xy,  u  is  an  explicit  function  of  the  ar-guments  x  and  y, 
X  is  an  implicit  function  of  the  arguments  u,  y;  y  is  an  implicit  function 
of  the  arguments  u,  x. 

Again,  if  the  letters  x,  y,  z  be  related  to  each  other  by  means  of  such 

an  equation  as  2^  +  xy^  +  4  yz  +  5  a:^  =  0, 

then  each  letter  is  an  implicit  function  of  the  other  two. 


3-4.]  INTRODUCTION  6 

An  explicit  function  of  one  or  more  numbers  is  known, 
given,  or  determined  in  terms  of  those  numbers.  It  is  sym- 
metric, algebraic,  rational,  transcendental,  etc.,  according 
as  the  expression  which  gives  it  its  value  is  symmetric, 
algebraic,  etc. 

E.g.,  if  y  = ,  then  y  is  irrational  in  b,  symmetric  as  to  m  and 

n,  transcendental  as  to  x,  algebraic  as  to  m,  n,  a,  b,  also  as  to  c  when  x  is 
commeusui'able. 

If  one  number  (or  function)  depends  on  its  arguments  in 
the  same  way  as  another  number  depends  on  its  arguments ; 
i.e.,  if  the  expressions  involved  are  of  the  same  form,  then 
the  first  number  is  said  to  be  the  same  function  of  its  argu- 
ments as  the  second  number  is  of  its  arguments. 

E.g.,  if  ax^  +  bx  =  c,    dy^  ■\-  ey  =f, 

then  c  is  the  same  explicit  function  of  a,  b,  x  as  /  is  of  d,  e,  y;  and  x  is 
the  same  implicit  function  oi  a,  b,  c  as,  y  is  of  </,  e,f. 

A  function  may  be  denoted  by  any  convenient  letter  or 
symbol,  as  /,  jP,  <^,  •••  with  or  without  indices,  and  followed 
by  the  argument,  inclosed  in  parentheses.  During  any 
investigation  the  same  functional  symbol  will  stand  for  the 
same  operation  or  series  of  operations. 

E.g.,  if  /(x)  =  a;»  -  az,    then  f{y)  =  y^  -  ay,  f(xy)  =  xV  -  a^ry. 

If  </)  (a;,  3/)  =  0  («/,  a;),  then  (Art.  3)  <f>  is  a  symmetric  func- 
tion of  X  and  y. 

If  y  =  F(x),  X  is  often  denoted  by  F-^(i/),  and  is  called 
the  inverse  ^-function  of  y.  This  notation  is  illustrated  in 
connection  with  "inverse  operations"  in  Art.  2. 

EXERCISES 

1.  If  f(x,  y)  =  ax^  +  bxy  +  cy^  write  f{y,  x) ;  /(x,  x) ;  f(,y,  y). 

2.  What  relation  must  exist  between  the  coeflBcients  in  exercise  1  to 
make  it  a  symmetric  function  ? 


6  DIFFERENTIAL   CALCULUS  [1st. 

3.  If  <^  (x-,  </)  =  4  J7/  f  x2  +  z  +  4  .y  -  7,  show  that  <^  (1,  2)  =  <^  (2,  1 ). 
Does  this  prove  <^  (r,  y)  to  be  a  syninuitric  function? 

4.  Let  i/(  (r,  »/)  =  .1  x  -\-  By  +  C.     Show  that  i//  (x,  y)  —  i),  xp  {y,  —  a:)  =  0 
are  the  equations  of  two  perpendicular  lines. 

What  curves  are  represented  by 

5.  If  /(x)  =:2a:Vl  —  x"^,   show   that  /(sin -]  =sina;. 
Find  the  value  of  /(cos^V 

6.  What  functions  satisfy  the  relations 

«^(x  +  //)  =  <^  (.r)  •  ^  (y)V  exponential  functions. 

/(^)  +  /(//)  =  y(-i'^)  '•■'  logarithms. 


^  (2 x)  =  2 \l/  (x)  \/nr[^(x)]2 ?    sine. 
X  ("2  a:)  =       ~rrM2 '  tangent. 

7.  If /(x)  =  x2  +  3,  and  F(x)=  2  -  Vi,  find/[F(x)],  and  F[/(x)]- 

8.  In  the  last  example,  find   /"[/(x)]  or/2(x),  /•'[F(x)]  or  F2(x), 
/-i(x),  F-Hx). 

9.  With  the  same  notation,  calculate  P  +  2/F  -  :5  F^. 
10.    If  <f>(x)  =  ^^-^,  show  that 


x+  1  1  +  (f>{.r)-<l>(y)      1  +  xy 

11.  Given    fix)  =  log  f^:^,  show  that  /(x)  +/(*/)  =/(f±X^  . 

12.  If .;  (^x)  =  VI  -  x2,  what  is  /(Vl  -  x'^)  ? 
Does  it  follow  that  if  <^(x)  =  y,  <f>{y)  =  x'i 

Give  examples  of  cases  in  which  this  is  not  true ;  in  which  it  is  true. 

13.  If  f{xy)  =f(x)  +f(y),  prove  that  /(I)  =  0. 

14.  If /(x  -f  .V)  =  /(x)  +  /(?/),  show  that/(0)  =  0,  and  pf(x)  =  /(px), 
where  p  is  any  positive  integer. 

15.  Using  the  same  function  as  in   the  last  example,   prove   that 
/(mx)  =  mf(x),  where  m  is  any  rational  fraction. 

16.  If  ^  =  logt  (x  +  Vl  +  r2),  express  x  as  a  function  of  y. 


4-«.]  INTRODUCTION  .  7 

17.  (iiveii/(x)=  fi%  find/(«),/(l), /(O);  show  that  ia  this  iuwc- 

18.  Given  xi/  —  2x  +  y  =  n;  show  that  y  is  not  a  function  of  x  when 
n  =  2. 

2z  —  1 

19.  U  y  =  (ft(x)  = -,  show  that  x  =  <f>  {y),  and  that  x  =  <;^'^(a:). 

8  J  —  2 

20.  If  ?/  =f(x)  ■=  and  3  =f(y),  find  z  as  a  function  of  a:. 

1  —  X 


5.  Constants  and  variables.  Usually,  during  an  investiga- 
tion, some  of  the  numbers  that  enter  into  it  preserve  their 
values  unchanged  ;  while  all  the  other  numbers  take  a  series 
of  different  values. 

A  constant  number  is  one  that  always  remains  the  same 
througliout  the  investigation. 

A  variable  number  is  one  that  changes  its  value,  so  that 
at  different  stages  it  requires  different  numerals  to  express  it. 

The  word  number  will  usually  be  omitted,  and  the  words 
constant  and  variable  will  be  used  alone,  in  problems  where 
it  is  necessary  to  distinguish  between  them. 

If  y  be  expressed  in  terms  of  x  by  the  relation  y  =  <^(a;), 
then,  if  a  numerical  value  be  given  to  x,  the  corresponding 
value  of  y  may  be  computed  ;  and  if  another  value  be  given 
to  x,  a  new  value  can  be  found  for  y,  and  so  on.  In  this 
equation,  both  x  and  y  are  varial)les,  but  if  the  vahie  of  x  at 
any  instant  be  given,  the  resulting  value  of  y  is  known.  In 
such  a  case,  x  is  called  the  independent  variable,  and  y  the 
dependent  variable.  The  argument  is  the  independent  varia- 
ble, the  function  is  the  dependent  one. 

6.  Continuous  variable ;  continuous  function.  When  the 
variable,  in  passing  from  one  value  to  another,  passes  through 
every  inteimediate  value  in  order,  then  it  is  continuous. 


8  DIFFERENTIAL   CALCULUS  [Int.  6. 

A  function  f(x~)  of  a  continuous  variable  x  is  called  a  con- 
tinuous function  in  the  interval  from  x  =  a  to  x  =  b,  ii  it  lias 
the  following  properties  : 

It  remains  real  and  finite  when  x  takes  any  real  value 
in  the  assigned  interval. 

For  each  value  of  x,  the  function  has  either  a  single 
value  or  any  number  of  determinate  values. 

As  X  changes  from  m  to  n  (two  arbitrary  numbers  within 
the  interval),  the  function  f(x),  if  single-valued, 
changes  from  /(m)  to  f(n)  by  passing  through  every 
intermediate  value,  in  order,  at  least  once ;  and,  if 
/(a;)  is  multiple-valued,  each  value  of  f(x)  changes 
from  a  particular  value  oif(m)  to  a  corresponding  par- 
ticular value  of  f(ri)i  in  such  a  way  as  to  pass  through 
every  intermediate  value,  in  order,  at  least  once. 

If,  at  a  value  x=  h,  any  one  of  these  conditions  fail,  the 
function  is  said  to  have  a  discontinuity  at  x  =  h. 

The  increment  taken  by  a  variable,  in  passing  from  one 
value  to  another,  is  the  difference  obtained  by  subtracting 
the  first  value  from  the  second.  An  increment  of  x  will  be 
expressed  by  the  symbol  Ax. 

It  is  implied  in  the  definition  of  a  continuous  function 
that  for  any  small  increment  of  the  variable,  the  increment 
of  the  function  is  also  small,  and  that  to  the  variable  an  in- 
crement can  always  be  given,  so  small  that  the  correspond- 
ing increment  of  the  function  shall  be  smaller  than  any 
number  that  may  be  assigned,  no  matter  how  small. 

E.g.,  if  y  =  f(x)  is  a  continuous  function  of  x  in  the  vicinity  of  the 
value  X  =  x„  then  corresponding  to  any  number  c  previously  assigned, 
another  number  8  can  be  assigned,  such  that  when  Ax  remains  numeri- 
cally less  than  8,  then     Ay  =/(a:,  +  Ax)  — /(ar,) 
shall  remaui  numerically  less  than  e.     (For  illustrations  see  Art.  14.) 


CHAPTER   I 

FUNDAMENTAL   PRINCIPLES 

This  chapter  treats  of  the  fundamental  ideas  of  a  limit 
and  of  an  infinitesimal,  and  uses  them  to  lead  up  to  the 
notion  of  a  derivative,  with  which  the  Calculus  is  so  largely 
concerned. 

7.  Limit  of  a  variable.  If  a  variable  take  successive 
values  that  approach  nearer  and  nearer  to  a  given  con- 
stant, so  that  the  difference  between  the  variable  and  the 
constant  may  become  smaller  than  any  number  that  can  be 
assigned,  then  the  constant  is  called  the  limit  of  the  variable. 

This  definition  applies  whether  the  variable  be  always 
greater  or  always  less,  or  sometimes  greater  and  sometimes 
less  than  the  constant. 

E.g.,  the  circumference  of  a  circle  is  the  limit  of  the  perimeter  of  an 
inscribed  polygon,  and  also  the  limit  of  the  perimeter  of  a  circumscribed 
polygon  when  the  length  of  the  sides  is  made  less  than  any  assigned 
number.  Similarly  the  area  of  the  circle  is  the  common  limit  of  the 
areas  of  the  inscribed  and  circumscribed  polygons. 

The  slope  of  a  tangent  to  a  curve  is  the  limit  of  the  slope  of  a  secant, 
when  two  points  of  intersection  of  the  secant  with  the  curve  approach 
coincidence. 

Thus  far  the  illustrations  apply  to  either  the  first  or  second 
case,  in  which  the  variable  is  either  always  less  or  always 
greater  than  its  limit.  An  illustration  of  the  third  case, 
wherein  the  variable  may  be  sometimes  greater  and  sometimes 

9 


10  DIFFERENTIAL   CALCULUS  [Ch.  I. 

less  than  the  constant,  is  furnished  by  a  decreasing  geometric 
progression  with  a  negative  common  ratio. 

For  instance,  consider  the  series  1,  —  J,  +  J,  —  J,  •••,  in  which  the 
number  of  terms  is  infinite.  The  sum  of  n  terms  of  this  series 
approaches  |  as  a  limit,  when  n  is  taken  larger  and  larger.  The  first 
term  is  1 ;  the  sum  of  the  first  two  terms  is  I ;  tlie  sum  of  the  first  three 
is  f;  of  the  first  four,  f;  and  so  on;  and  these  successive  sums  ai'e 
alternately  greater  and  less' than  |,  but  any  one  of  thein  is  nearer  f  than 
any  sum  preceding.  By  taking  terms  enough,  a  sum  can  be  reached  that 
will  differ  from  f  by  less  than  any  number  that  may  be  assigned ;  for  the 
sum  of  n  terms  of  this  geometric  series  is 

hence  s„-|  =  -2(-  ^)", 

which  can  evidently  be  made  less  than  any  assigned  number  by  sufficiently 
increasing  n. 

8.  Infinitesimals  and  infinites.  A  variable  that  approaches 
zero  as  a  limit  is  an  infinitesimal.  In  otlier  words,  an  infini- 
tesimal is  a  variable  that  becomes  smaller  than  any  number 
that  can  be  assigned. 

The  reciprocal  of  an  infinitesimal  is  then  a  variable  that 
becomes  larger  than  any  number  that  can  be  assigned,  and 
is  called  an  infinite  variable. 

E.f/.,  the  number  (^)'»  which  presents  itself  in  the  last  illustration  is 
an  infinitesimal  when  ti  is  taken  larger  and  larger;  and  it's  reciprocal 
2"  is  an  infinite  variable. 

From  these  definitions  of  the  words  "  limit "  and  "  infini- 
tesimal "  the  following  useful  corollaries  are  immediate 
inferences. 

Cor.  1.  "Tiie  difference  between  a  variable  and  its  limit 
is  an  infinitesimal  variable. 

Cor.  2.  Conversely,  if  the  difference,  between  a  constant 
and  a  variable  be  an  infinitesimal,  then  the  constant  is  the 
limit  of  the  variable. 


7-9.3  FUNDAMENTAL   PlilNt'IPLES  11 

For  convenience,  the  symbol  =  will  be  placed  between 
a  variable  and  a  constant  to  indicate  that  the  variable 
approaches  the  constant  as  a  limit ;  thus  the  symbolic  form 
x  =  a  is  to  be  read  "  the  variable  x  approaches  the  constant  a 
as  a  limit.'' 

The  corollaries  just  mentioned  may  accordingly  be  sym- 
bolically stated  thus : 

1.  \l  X  =  a,  then  a;  =  a  +  «,  wherein  «  =  0 ; 

2.  li  X  =  a  -\-  a,    and  «  =  0,  then  x  =  a. 

It  will  appear  that  the  chief  use  of  Cor.  1  is  to  convert 
given  "  limit  relations  "  into  the  form  of  ordinary  equations, 
80  that  they  may  be  at  once  combined  or  transformed  by  the 
laws  governing  the  equality  of  numbers ;  and  then  Cor.  2 
will  serve  to  express  the  final  result  in  the  original  form  of  a 
limit-relation. 

In  all  cases,  whether  a  variable  actually  becomes  equal  to 
its  limit  or  not,  the  important  property  is  that  tlieir  difference 
is  an  infinitesimal.  An  infinitesimal  is  not  necessarily  in  all 
stages  of  its  history  a  small  number.  Its  essence  lies  in  its 
power  of  decreasing  numerically,  having  zero  for  its  limit, 
and  not  in  the  smallness  of  any  of  the  constant  values  it  may 
pass  through.  It  is  frequently  defined  as  an  "infinitely 
small  quantity,"  but  this  expression  should  be  interpreted  in 
the  above  sense.  Thus  a  constant  number,  however  small  it 
may  be,  is  not  an  infinitesimal. 

9.  Fundamental  theorems  concerning  infinitesimals,  and 
limits  in  general.  This  article  will  be  devoted  to  a  rigorous 
treatment  of  the  theory  of  limits  so  far  as  necessary  to  furnish 
a  logical  basis  for  the  process  of  differentiation  to  which 
the  chapter  leads  up.  Theorems  1-3,  which  are  special 
theorems   relating    to   infinitesimal    variables,   are   deduced 


12  DIFFERENTIAL   CALCULUS  [Ch.  I. 

immediately  from  tlie  definition  of  an  infinitesimal ;  and  are 
then  used  in  conjunction  with  the  corollaries  of  Art.  8,  to 
establish  the  general  theorems  4-9  relating  to  the  limits  of 
any  variables. 

Theorem  1.  The  product  of  an  infinitesimal  a  by  any 
finite  constant,  k,  is  an  infinitesimal;  i.e.,  if 

«  =  0, 

then  ka  =  0. 

For,  let  e  be  any  assigned  number ;  then,  by  hypothesis,  « 

can  become  less  than  -  ;  hence  ka  can  become  less  than  c,  the 
k 

arbitrary,  assigned  number,  and  is,  therefore,  infinitesimal. 

Theorem  2.  The  algebraic  sum  of  any  finite  number  (w) 
of  infinitesimals  is  an  infinitesimal ;  i.e.,  if 

a=0,  /3  =  0,   ..., 
then  a  +  /8  +  •••  =  0. 

For  the  sum  of  the  n  variables  does  not  numerically  ex- 
ceed n  times  the  largest  of  them,  but  this  product  is  an 
infinitesimal  by  theorem  1 ;  hence  the  sum  of  the  n  variables 
is  an  infinitesimal. 

Note.  The  sum  of  an  infinite  number  of  infinitesimals 
may  be  infinitesimal,  finite,  or  infinite,  according  to  circum- 
stances. 

E.g.,  let  a  be  a  finite  constant,  and  let  n  be  a  variable  that  becomes 
infinite;  then  _,  _,  .    ,  are  all  infinitesimal  variables;  but 


-"  I 

"'    "    n¥ 

—--\ h  •••  to  n  terms 

=  -  ,  which  is  infinitesimal, 
n 

while 

-  +  -  +  .••  to  n  terms 
n      n 

=  a,  which  is  finite, 

and 

A  +  ^  4-  ...  to  n  terras 

=  an^,  which  is  infinite. 

9.]  FUNDAMENTAL  PRINCIPLES  13 

Theorem  3.  The  product  of  two  infinitesimal  variables 
is  an  infinitesimal ;  i.e.,  if 

«  =  0,  /9  =  0, 

then  ayS  =  0. 

For,  let  c  be  any  assigned  number  <  1 ;  then  a,  ^,  can 
each  become  less  than  e ;  hence  «/3  can  become  less  than  e^y 
which  is  less  than  c,  since  c  <  1 ;  thus  «y8  can  become  less 
than  any  assigned  number,  and  is,  therefore,  infinitesimal. 

Note.     From  theorems  1-3,  it  follows  that,  if 
a  =  0,  /3  =  0,  7  =  0, 
then      aa  +  h^  -{-  cy  +  d^y  +  eya  +  /«/S  +  ga^y  =  0 
when  a,  b,  c,  c?,  e,  /,  g,  are  finite  constants. 

Theorem  4.  If  two  variables  {x,  y')  be  always  equal, 
and  if  one  of  them  (a:)  approach  a  limit  (a),  then  the  other 
approaches  the  same  limit ;  i.e..,  if 

X  =  y,  and  re  =  a, 
then  y  =  C" 

For,  by  Art.  8,  Cor.  1, 

X  =  a  -\-  a^ 
in  which  a  =  0  ; 

hence  y  =  a  +  a\ 

and,  therefore,  ^  =  «» 

by  Art.  8,  Cor.  2. 

Theorem  5.     The  limit  of  the  sum  of  a  constant  (<?)  and 

a  variable  (x)  is  equal  to  the  constant  plus  the  limit  of  the 

variable  ;  i.e., 

lim  (tf  +  a;)  =  c  +  lim  x. 

For,  let  x^a-. 


14  DIFFERENTIAL   CALCULUS  [On.  I. 

then  X  =  a  +  a^                        [Art.  8,  Cur.  1. 

in  which  .           a  =  0  ; 

therefore  c  +  x  =  c-\-a-\-a, 

hence  c  -\-  x  =  c  -\-  a,                 [Art.  8,  Cor.  2. 

i.e.,  lim  (c  +  x)=  c  +  a  =  <;  -\-  lim x. 

Theorem  6.*   The  limit  of  the  product  of  a  constant  and 

a  variable  is  equal  to  the  constant  multiplied  by  the  limit  of 

the  variable ;  i.e., 

limcr  =  ^lima:. 

For,  using  the  notation  of  theorem  5,  and  multiplying  by  <?, 

ex  =  ca  -{■  ea  \ 

therefore  ex  =  ca,  [Art.  8,  Cor.  2. 

i.e.,  ^       lim  ex  =  ea  =  dim  x. 

Theorem  7.  If  the  sura  of  a  finite  number. of  variables 
{x,y, ...)  be  variable,  then  the  limit  of  their  sum  is  equal  to 
the  sum  of  their  limits  ;  i.e., 

lim(a;  4-  ?/  +  •••)=  lima;  +  lim  «/  +  .-.. 

For,  let  X  =  a,  y  =  h,  ...  ; 

then  X  =  a  +  a,  y  =  b  +  ^,  ...,       [Art.  8,  Cor.  1. 

Avherein  «  =  0,  ^  =  0,  ...  ; 

hence     re  +  y +  ...=  (a  +  ft +  •••)  +  (« +j8  +  •••); 
but  ■    «  +  y8  +  ...  =  0, 

by  theorem  2 ;  hence,  by  Art.  8.  Cor.  2, 

limCa;  +  y  +  •••)=  a  +  h  +  ."  =  lima;  +  limy  +  •••• 
Note.     The  limit  of  the  sum  of  an  infinite  number  of 
variables  may  not  be  equal  to  the  sum  of  their  limits.     (Cf. 
Th.  2,  Note.) 

;  O).  the  sum  of  thfi  limits  of  — k      ,  ,       , 

„2       22         7^2       03         „2 


E.g.,  when  n  =  x,  the  sum  of  the  limits  of  -  H ,    —  +  — , h  — , 

•'  '  O    '    „2'      92        „-2       03         „i 


—  -\ — -.  is  1 ;  but  the  limit  of  their  sum  is  li. 

Qn        «■■' 


9.]  FUNDAMENTAL   PRINCIPLES  15 

Cou.     If  the  sum  of  a  Hnite  number  of  variables  (x^y^  z. ...) 

be  constant,  then  this  constant  (c?)  is  equal  to  the  sum  of  their 

limits  ;  i.e.,  if 

X-+  1/  +  z  -{-'■■  =  e, 

then  lim^'  +  Vun  t/  +  limz  +  •••  =  c. 

For,  by  transposition, 

t/  +  z  +  '■■  =  c  —  x; 

hence,  by  theorems  4,  7,  and  5, 

limy  4-  lim  s  -\-  •••  =  lim  (c  —  x)=  c  —  lima:; 

therefore       lim x  +  lim  9/  +  \\mz  +  •••  =  c.     [Cf.  Ex.  2,  p.  49. 

Theokem  8.  If  the  product  of  a  finite  number  of  variables 
(x,  y,  r,  •••)  be  variable,  then  the  limit  of  their  product  is 
equal  to  the  product  of  their  limits  ;  z.e., 

Yimxi/z  ...  =  lima:limy  lim  2  •••. 

For,  using  the  previous  notation,  and  taking  the  case  of 
three  variables, 

xi/z  =(a  +  u)(b  +  ^~)(c  +  7) 

=  ahc  +  aby  -\-  hca  +  ca^  +  hwy  +  ea^  +  a/87  +  «)^7  '■> 
hence",  by  theorems  1,  2,  8,  and  Art.  8,  Cor.  2, 
lim  xyz  =  ahc  =  lim  x  lim  y  lim  z. 

CoK.  If  the  product  of  a  finite  number  of  variables 
(a:,  y,  2)  be  constant,  then  this  constant  is  equal  to  the  prod- 
uct of  their  limits  ;  i.e.^  if  xyz  =  c,  then  lim  x  lim  y  lim  z  =  e. 

For,  let  w  be  any  other  variable,  then 

xyzw  =  cw; 
hence,  by  theorems  4,  6,  8, 

lim  X  lim  y  lim  z\imw  =  c  lim  w, 
therefore  lim  x  lim  y\imz  =  .?.     [Cf.  Ex.  2,  p.  49. 


16  DIFFERENTIAL   CALCULUS  [Ch.  1. 

Theorem  9.  If  the  quotient  of  two  variables  (a;,  y)  be 
variable,  then  the  limit  of  their  quotient  is  equal  to  the 
quotient  of  their  limits,  provided  these  limits  are  finite ; 

X      lim  X 


I.e., 


lim-  = 


1/      lim  y 


For,  since  x=  -•  y,  hence  by  theorems  4,  8, 
lim  X  =  lim  -  •  lim  y; 

y 

^1        «  ■,.     X      lima; 

therefore  lim  -  = -• 

y      liin  y 

Cor.  1.  If  the  quotient  of  two  variables  (x,  ,y)  be  con- 
stant, then  this  constant  (c)  is  equal  to  the  quotient  of  their 
limits; 

■  c                         X          ,1        lim  X 
I.e..,  II  -=  Ci  then =  c. 

y  lim^/ 

For,  since  x  =  cy,  hence  by  theorems  4,  6, 

lim  x  =  c  lim  y, 

,1        o  lim  X 

therefore  r: =  c. 

ivaiy 

Cor.  2.  The  limit  of  the  quotient  of  a  constant  (c)  and 
a  variable  (x)  is  equal  to  the  constant  divided  by  the  limit 
of  the  variable; 

I.e.,  lim-  = 


X      lima; 
For,  let  -  =  y.,  then  c  =  xy,  and  by  theorem  8,  Cor., 

X 

e  =  lim  X  limy; 

therefore      lim  y  = -;  that  is,  lim  -  = . 

lima;  x     lima; 


9-10]    ,  FUNDAMENTAL   PRINCIPLES  17 

10.  Comparison  of  variables.  Some  of  the  principles  just 
esttiblished  will  now  be  used  in  comparing  variables  with 
each  other.  The  relative  importance  of  two  variables  that 
are  approaching  limits  is  measured  by  the  limit  of  their 
ratio. 

Debmnition.  One  variable  (a)  is  said  to  be  infinitesimal, 
infinite,  or  finite,  in  comparison  with  another  variable  (a;) 
when  the  limit  of  their  ratio  (a :  x)  is  zero,  infinite,  or  finite. 

In  the  first  two  cases,  the  phrase  "infinitesimal  or  infinite 
in  comparison  with  "  is  sometimes  replaced  by  the  less  pre- 
cise phrase  "infinitely  smaller  or  infinitely  larger  than."  In 
the  third  case,  the  variables  will  be  said  to  be  of  the  same 
order  of  magnitude. 

The  following  theorem  and  corollary  are  useful  in  com- 
paring two  variables : 

Theorem  10.  The  limit  of  the  quotient  of  any  two 
variables  (a;,  y)  is  not  altered  by  adding  to  them  any  two 
numbers  (a,  y8),  which  are  respectively  infinitesimal  in 
comparison  with  these  variables  {x,  y); 

1 .     X  +  a      1  •     X 
I.e.,  lim  — ——  =  lim  -, 

y  +  py 

provided  -  =  0,^:^0. 


For,  since 


X  ■\-  a 


l  +  « 


hence,  by  theorems  4,  8, 


+ 

y 


1  +  -? 

,.     a;  +  «      ,.     X    ..             X 
lim ;=  =  hm  -  •  lim r^ ; 

^+^       y      1+^ 


18  DIFFERENTIAL   CALCULUS  [Ch.  I. 

but,  by  theorems  9,  5,  and  hypothesis, 

lira =  1 ; 

y 

therefore,  lim  — — ^  =  Hm  — 

y  +  p        y 

Coii.  If  the  difference  (S)  between  two  variables  {x^  y) 
be  infinitesimal  as  to  either,  tlie  limit  of  their  ratio  is  1,  and 
conversely ; 

I.e.,  if  ^  =  0,  then-  =  l. 

y  y 

For,  since  x  —  y  =  8,  then  a*  =  y  +  8, 


and  lim  -  =  lim  ^^^-^  =  lim    1  +  "  =  1. 

y  y  \     yJ 

Conversely,  if  =1,  then ^  =  0. 

y  y 

For,  by  Art.  8,  Cor.  1, 

--1=0;  z.g.,^i:ii^=0. 

y  y 

11.  Comparison  of  infinitesimals,  and  of  infinites.  Orders 
of  magnitude.  It  has  already  l)een  stated  that  any  two 
variables  are  said  to  be  of  the  same  order  of  magnitude 
when  the  limit  of  their  ratio  is  a  finite  number;  that  is  to 
say,  is  neither  infinite  nor  zero.  Tn  less  precise  language, 
two  variables  are  of  the  same  order  of  magnitude  when 
one  variable  is  neither  infinitely  larger  nor  infinitely  smaller 
than  the  other.  For  instance,  k^  is  of  the  same  order  as  ^ 
when  k  is  any  finite  number;    thus  a  finite  multiplier  or 


10-11.]  FUNDAMENTAL    PRINCIPLES  19 

divisor  does  not  affect  the  order  of  magnitude  of  any 
variable,  whether  infinitesimal,  finite,  or  infinite. 

In  a  problem  involving  infinitesimals,  any  one  of  them,  «, 
may  be  chosen  as  a  standard  of  comparison  as  to  magnitude; 
then  a  is  called  the  principal  infinitesimal  of  the  first  order 
of  smallness,  and  its  reciprocal  a~^  is  the  principal  infinite  of 
the  first  order  of  largeness ;  u^  is  called  the  principal  infini- 
tesimal of  the  second  order  of  smallness,  and  its  reciprocal 
a~''  is  the  principal  infinite  of  the  second  order  of  largeness. 
In  general,  «"  is  called  the  principal  infinitesimal  of  order  n, 
when  n  is  either  a  positive  integer  or  a  positive  fraction; 
but  when  n  is  any  negative  number,  a"  is  the  principal 
infinite  of  the  corresponding  positive  order  (  — w).  An 
infinitesimal  or  infinite  of  order  zero  is  a  finite  number. 

Besides  the  principal  infinitesimal  («")  of  the  wth  order, 
there  are  many  other  infinitesimals  of  the  same  order  of 
smallness,  for  instance,  any  infinitesimal  of  the  form  ka", 
in  which  k  is  any  finite  multiplier. 

To  test  for  the  order  (w)  of  any  given  infinitesimal  (yS) 
with  reference  to  the  standard  infinitesimal  («)  on  which 
it  depends,  it  is  necessary  to  select  an  exponent  n,  such  that 

™rt  —  =  k,  some  finite  constant. 

a  =  0  ^« 

E.g.,  to  find  the  order  of  the  variable  Zx*  —  4x%  with  reference  to 
t  as  the  base  infinitesimal. 

Comparing  with  x^,  x^,  x*,  in  succession  : 

xTo  ^^^^r^  =  xTo  (^ ^'  -  4 x)  =  0,  not  finite ; 

lim  3  x^  -  4  x8       lini   ^3  ^  _  4)  ^  _  4^  ^^^^6 ; 


0  3^8  x  =  0 

—  ix^       lim   /q      4\  ,  /J   .4. 


lim  3  X*  —  4  x^ 

x  =  0 


hence  3x*  —  ix^  is  an  infinitesimal  of  the  same  order  of  smallness  as  x'; 
that  is,  of  the  tliird  order. 

DIFF.   CALC.  — 3 


20  DIFFERENTIAL    CALCULUS  [Ch.  I. 

The  order  of  largeness  of  an  infinite  variable  can  be 
tested  in  a  similar  way.  For  instance,  if  x  be  taken  as  the 
base  infinite,  let  it  be  required  to  find  the  order  of  the  variable 
'6  3^  —  4:3^.     Comparing  with  a^  and  a^  : 

lirn    3^4__4^^    li^    (3a:-4)=a); 


hence  82;*  — 4  a^  is  an  infinite  of  the  same  order  of  large- 
ness as  2^,  that  is,  of  the  fourth  order. 

The  process  of  finding  the  limit  of  the  ratio  of  two  in- 
finitesimals is  facilitated  by  the  following  principle,  based 
on  theorem  10  of  Art.  10 :  The  limit  of  the  quotient  of  two 
infinitesimals  is  not  altered  by  adding  to  them  (or  subtract- 
ing from  them)  any  two  infinitesimals  of  higher  order,  re- 
spectively. 

lira     3ar2  4  x*        lim  Sx^     3 


E.g., 


''  =  ^4z^-2xi     ^  =  ^4x2     4 


This  principle  is  sometimes  called  "  the  fundamental  theo- 
rem of  the  Differential  Calculus,"  owing  to  its  use  in  the 
"  fundamental  problem "  stated  in  Art.  15. 

12.  Order  of  magnitude  of  expressions  involving  infinitesi- 
mals or  infinites. 

Theorem  11.  The  product  of  two  infinitesimals  is  another 
infinitesimal  whose  order  is  the  sum  of  their  orders. 

For,  let  j3,  7  be  infinitesimals  of  orders  m,  n,  with  refer- 
ence to  the  base  infinitesimal  a;  then,  by  definition, 

li."l  ^  =  6,     ^V"  ^=.0,  where  b,  c  are  finite ; 


11-12.]  FUNDAMENTAL  PRINCIPLES  21 

hence,  multiplying  and  using  theorem  8, 

im    P7^  _  J     ^  finite  number, 

therefore  ^y  is  an  infinitesimal  of  order  m  +  n. 

Cor.  1.  The  product  of  two  infinites  is  another  infinite 
whose  order  is  the  siim  of  their  orders. 

For,  let  /3,  y  be  infinites  of  orders  ?w,  n ;  then,  since  the 
principal  infinites  of  these  orders  are  «"",  a~", 

lim  _^  ^  J       lim  _7_  _  ^ 

therefore  V",,     ^^     =  ?»<?,  a  finite  number  : 

hence  /Sy  is  of  the  same  order  as  a"^""^"',  that  is,  an  infinite 
of  order  m  +  n. 

Cor.  2.  The  product  of  an  infinitesimal  of  order  m  by 
an  infinite  of  order  7i  is  an  infinitesimal  of  order  m  —  n  when 
m>n;  but  it  is  an  infinite  of  order  n  —  m  when  n>m. 

For,  since  ^'""A^h,     ^'^^=0; 

hence  I\«l-^  =  J(,=  "."^.-f^' 

therefore  when  m>n^  fiy  is  an  infinitesimal  of  order  w  —  w, 
and  when  n>m,  ^y  is  an  infinite  of  order  n  —  m. 

Theorem  12.  The  quotient  of  an  infinitesimal  of  order 
m  by  an  infinitesimal  of  order  n  is  an  infinitesimal  of  order 
m  —  n  when  m>n;  but  it  is  an  infinite  of  order  n  —  m 
when  n>m. 


For,  since  ^™n^=J,     ^™n^  =  e. 


a™  ".  -  "  a" 


22  DIFFERENTIAL    CALCULUS  [Cii.  1. 

therefore,  dividing  and  using  theorem  9, 

lim  _7__^_    lini        7      . 
a  =  0  „"•-«  "~  ^,  ~  a  =  0  «-(»-"•) ' 

hence  —  is  an  intinitesimal  of  order  m  —  w.,  when  wi  >  w,  and 

7     ,    . 
it  is  an  infinite  of  order  n  —  m,  when  w  >  m. 

Cor.  1.  The  quotient  of  an  infinite  of  order  m  by  an 
infinite  of  order  n  is  an  infinite  of  order  m  —  w,  when  m  >  n ; 
but  it  is  an  infinitesimal  of  order  w  —  m  when  n  >  m. 

Cor.  2.  The  ratio  of  two  infinitesimals  is  finite,  infini- 
tesimal, or  infinite  according  as  the  antecedent  is  of  the 
same  order,  a  higher  order,  or  a  lower  order,  than  the  conse- 
quent. 

Cor.  3.  The  ratio  of  two  infinites  is  finite,  infinitesimal, 
or  infinite  according  as  the  antecedent  is  of  the  same  order, 
a  lower  order,  or  a  higher  order,  than  the  consequent. 

Theorem  13.  The  order  of  an  infinitesimal  is  not  altered 
by  adding  or  subtracting  another  infinitesimal  of  higher 
order. 

For,  let  /8,  7  be  two  infinitesimals  of  order  w,  w,  in  which 

w  <  M,  then 

lim    ^  _.  j^     lim    '!_  —  (, 
0  =  0  «'«  ~    '  a  =  0  a"         ' 

hence  lim       „  ^  =  lim  —-  +  lim  -^1 

«"»  a*"  a'" 

but  -^  is  an  infinitesimal  of  order  n  —  7/1,  by  theor.  12; 


a 


7 


thus  lim  -^  =  0, 


12.]  FUNDAMENTAL  PRINCIPLES  23 

and  lim  — — ^  =  hin  —-  =  6, 

therefore  /S  +  7  is  an  intinitesimal  of  the  same  order  as  /8. 

Note.  The  order  of  an  infinitesimal  is  not  altered  by 
adding,  but  may  be  altered  by  subtracting,  another  infini- 
tesimal of  the  same  order  and  sign. 

For  instance,  let  y8  =  -5  «2  -f  4  «3,  of  second  order, 
y  =  :\  «2  _  2  «3^  of  second  order, 
then  /8  +  7  =  (J  «2  ^  2  «^,  of  second  order, 

but  /S  —  7  =  <)  «^  of  third  order. 

Cor.  The  sum  of  a  finite  number  of  infinitesimals  of  the 
same  sign  is  an  'infinitesimal  of  an  order  equal  to  the  lowest 
order  among  the  infinitesimals  summed. 

Theorem  14.  The  order  of  an  infinite  is  not  altered  by 
adding  or  subtracting  another  infinite  of  lower  order. 

Note.  The  order  of  an  infinite  is  not  altered  by  adding, 
but  may  be  altered  by  subtracting,  another  infinite  of  tlie 
same  order  and  sign.     (Proof  and  illustration  as  above.) 

Cor.  The  sum  of  a  finite  number  of  infinites  of  the 
same  sign  is  an  infinite  of  an  order  equal  to  the  highest 
order  among  the  infinites  summed. 

Theorem  15.  The  limit  of  the  finite  sum  of  any  num- 
ber of  infinitesimals  is  not  altered  by  replacing  any  infini- 
tesimal by  another  that  bears  to  it  a  ratio  whose  limit  is 
unity. 

For,  let  «i  +  «2  +  *"  +  "»•• 

be  the  sum  of  n  infinitesim  ils  of  such  a  nature  that  as  n  in- 


24  DIFFERENTIAL  CALCULUS  [Ch.  I. 

creases,  each  term  decreases  so  that  the  limit  of  the  sum  is 
finite. 

Let  there  be  n  other  infinitesimals, 

Hli    Hv    '"■>    Pnt 

SO  related  to  the  first  set  that 

lin3  =  1,  Hm^  =  1,  •••  lim^  =  1, 
then 
^l=l+e„f?=l+6„   ...^»=l  +  .„,    [e,  =  0,  e,  =  0... 

and     ^j  =  aj  +  e^a^,  ^^  =  a^  +  62^2'   "•  l^n  =  ««  +  f«««^ 
hence 

/3l  +  /32-| |-^«=(«l  +  «2+  •••  +«n)  +  (fl«l+.f2«2+  ••*  +f»««)- 

Next  let  7]  be  an  infinitesimal  that  is  numerically  equal  to 
the  largest  of  the  e's, 

then     eittj  +  e^a^  +  •••  +  e„a„  |<|  17  (aj  +  «2  +  **•  +  «»)»* 
hence 

(/Si+y82+".+y9„)-(ai  +  «2+-+«»)|<h(«i  +  «2+ •••+««)• 
Taking  limits  and  remembering  that,  by  hypothesis, 
lim  («j  +  ftg  +  •••  +  a„)  is  finite,  and  lim?;  =  0, 
it  follows  that 

lim  (/Si  +  /S2  +  -  +  ^n)  =  lim  («i  +  «2  +  •••  +  «„). 

Note.  This  theorem  may  sometimes  be  conveniently 
stated  as  follows :  the  limit  of  the  finite  sum  of  infinitesi- 
mals is  not  altered  if  these  infinitesimals  be  replaced  by 
others  which  differ  from  them  respectively  by  infinitesimals 
of  higher  order,  f 

*  The  symbol  (<|  stands  for  "  is  numerically  less  than."    (See  Art.  64.) 
t  This  is  called  the  "fundamental  theorem  of  the  Integral  Calculus." 


12-13.]  FUNDAMENTAL   PRINCIPLES  25 

13.   Useful  illustrations  of  infinitesimals  of  different  orders. 

rp^,_,__„--  1  liiu  sin  0       -,         lim  tan  0       ■• 

Theorem  1.      ^^q_— =  1;    ^^^___=i. 

With  0  as  a  center  and  OA  =  r  b  d 

as  radius,  describe  the  circular 
arc  AB.  Let  the  tangent  at  A 
meet   OB   produced  in   I);    draw 

O  ''  V  A 

BC  perpendicular  to  OA,  cutting  ^^^  ^ 

OA  in  0.  Let  the  angle  AOB  =  ^ 
in  radian  measure, 

then  arc  AB  =  r0, 

CB  <  arc  A5  <  AD,  by  geometry, 

I.e.,  r  sin  6  <r  6  <r  tan  ^, 

sin  6  <0  <  tan  ^. 

By  dividing  each  member  of  these  inequalities  by  sin  6^ 

1  <  -^  <  sec  0, 
sin  0 

but  when  ^  =  0,  sec  ^  =  1, 

hence  /."V  ^  =  1,  and    "."\  ^  =  1. 

Similarly,  by  dividing  the  inequalities  by  tan  0, 

cos  0  <  — ^  <  1, 
tan  0 

1  lim        0  1  J     lini    tan  0       ■, 

hence  /,  .  ^ =  1,  and  ^  .  f.  — — -  =  1 . 

^  =  ^  tan  0  ^  =  ^     0 

Cor.  1.     The  numbers  0,  sin  0,  tan  ^  are  infinitesimals  of 
the  same  order. 

Cor.  2.     The  expressions  sin  0  —  0,  tan  ^  —  ^  are  infini- 
tesimal as  to  0. 


26  DIFFERENTIAL    CALCULUS  [Ch.  I. 

Theorem  2.  If  one  angle  ^,  of  a  right  triangle,  be  an 
infinitesimal  of  the  first  order,  then  the  hypothenuse  r  and 

the  adjacent  side  x  are  either  both 
finite,  or  they  are  infinitesimals  of 
the  same  order ;  and  the  opposite 
side  y  is  an  infinitesimal  of  order 
^'^  ^  one  higher  than  that  of  r  and  x. 

For-=cos^,   which    approaches  the  value   1   as  ^  =  0 ; 
r 

hence  x,  r  are  infinitesimals  of  the  same  order;  which  may 

be  the  order  zero. 

Also  y  =  rsinO^ 

and  sin  0  is  of  order  1  ;   therefore  y  is  of  order  one  higher 
than  r. 

Cor.  In  the  same  case,  if  0  be  of  the  first  order,  and  if  r 
and  X  be  of  the  order  n,  then  the  difference  between  r  and  x 
is  an  infinitesimal  of  order  n  +  2. 

r^sin^^ 


For  r^  ~  x^  =  y^  =  r^  sin^  0,    r  —  x  = 


r  +  z 


but  the  orders  of  r^,  sin^^,  r  +  x,  are  respectively  2w,  2,  w ; 

.'.  r  —  a;  is  of  order 

2n  +  2  —  n  =  n  +  2. 

Theorem  3.  The  difference  between  the  length  of  an 
infinitesimal  arc  of  a  circle  and  its  chord  is  of  at  least  the 
third  order  when  the  arc  is  the  first  older. 

For,  let  CD  be  the  arc,  and  CB,  DB,  tangents  at  its  ex- 
tremities ;  then 

chord  CD  <  arc  CD  <  DB  +  BC. 

Let  the  angle  BOD  =  ^  be  taken  as  the  principal  infini- 
tesimal;  then,  since  -avcCD  =  2rd,  and  r  is  finite,  hence 
arc  CD  is  of  order  X, 


13.] 


FUNDAMENTAL   PRINCIPLES 


27 


Fig.  3. 


Again,  since  AD  is  of 
order  1  (Th.  1,  Cor.  1), 
and  angle  ADB  =  ^  is  of 
order  1,  hence  DB  is  of  (T 
order  1,  and  BB  —  BA  is 
of  order  3  (Tli.  2,  Cor.); 
.  • .  arc  CB  —  chord  OB  is 
of  order,  at  least,  three. 

Theouem  4.  The  difterenae  between  the  length  of  any 
infinitesimal  arc  (of  finite  curvature),  and  its  cliord,  is  an 
infinitesimal  of,  at  least,  the  third  order. 

Note.  The  curvature  is  said  to  be  finite  when  the  limit- 
ing ratio  of  the  length  of  a  small  chord  to  the  angle  between 
the  tangents  at  its  extremities  is  finite,  and  not  zero. 

I'll  us,  in  the  present  case,  the  cliord  PQ  and  the  angle 
TSP  are,  by  hypothesis,  infinitesimals  of  the  same  order.* 

Let  the  angle  TSF  be  the 
principal  infinitesimal ;  then, 
since 

TSF  =  S'QR  +  EPS, 

'^1  it  follows  that  the  greater  of 
Is  the  latter  two  angles,  say  RQS, 
is  of  the  first  order,  while  the 
other  may  be  of  the  first  or 
a  higher  order.  Also,  the 
greater  of  the  two  segments 
RQ,  PR,  say  the  latter,  is  of 
the  first  order,  while  RQ  may 
be  of  the  first  or  higher  order. 

*  If  TSP  wiiv  of  liiglter  order  than  PQ,  the  curvature  would  be  zero;  if 
of  lower  order,  the  curvature  would  be  infinite  ;  tiie  former  is  the  case  at  an 
inflection,  the  latter  at  a  cusp. 


Fin.  4. 


28  DIFFERENTIAL    CALCULUS  [Ch.  I. 

Again,  by  theorem  2,  QM,  QS  are  of  the  same  order,  and 
PR,  PS  are  of  the  same  order. 

Now       arc  QP  -  cliord  QP<QS+  SP  -  QP,        [geom. 

i.e.,  < ( QS  -  QM)  +  (>S'P  -  MP) ; 

but  since     QS  -  QR  =  QS(1  -  cos  /3)  =  2^^sin2|, 

and,  similarly,        SP-  RP=2SP  sin2  " 

and,  since  each  of  these  products  is,  at  least,  of  the  third 
order,  hence  arc  QP  —  chord  QP  is  of,  at  least,  the  third 
order. 

EXERCISES 

1.  Let  ABC  be  a  triangle  having  a  right  angle  at  C;  draw  CD  per- 
pendicular to  AB,  DE  [lerpendicular  to  CB,  EF  perpendicular  to  DB, 
FG  perpendicular  to  EB\  let  the  angle  BAC  be  an  infinitesimal  of  tlie 
first  order,  AB  remaining  finite.     Prove  that: 

CD,  CB  are  of  order  1 ; 

DB,  DE  are  of  order  2 ; 

EB,  EF,  {CB  -CD)  are  of  order  3; 

FB,  FG,  {DB  -  DE)  are  of  order  4. 

2.  Of  what  order  is  the  area  of  the  triangle  ABCl  BCD'i  CDEl 

3.  A  straight  line,  of  constant  length,  slides  between  two  rectangular 
straight  lines,  CAA',  CB'B;  let  AB,  A'B'  be  two  positions  of  the  line. 
Show  that,  in  the  limit,  when  the  two  positions  coincide, 

AA'  ^  CB 
BB'      CA ' 

14.  Continuity  of  functions.  From  the  foregoing  theorems 
on  limits,  and  the  definition  of  a  continuous  function,  the 
following  theorems  relating  to  continuity  are  easily  derived, 
and  applied  to  the  ordinary  classes  of  functions. 

Theorem  1.  If  a  variable  approach  a  constant,  as  a  limit, 
according  to  any  given  law,  then  any  function  of  the  variable 


13-14]  FUNDAMENTAL  PRINCIPLES  29 

approaches  the  same  function  of  the  constant  as  a  limit  if 
the  function  be  continuous  for  values  of  the  variable  in  the 
vicinity  of  the  constant. 

Let  f(x}  be  a  continuous  function  of  x,  for  values  of  x 
near  a ;  then  when  x  =  a  from  either  side 

/(a:)  =  /(a), 
regard  being  had  to  correspondence  of  multiple  values,  if  any. 

For,  let  ■  X  =  a  -{-  h, 

where  ^=0;  then       f  (a -\- K)  —  f  (a) 

can  be  made  less  than  any  assigned  number  from  the  defini- 
tion of  a  continuous  function  (Art.  6)  ;  hence 
/(a  +  70  =  /(a:)  =  /(a). 
Ex.   Prove  lira  /(x)  =/(lim  x),  i.e.,  the  operators  /,  Urn,  commutative. 

Cor.  Conversely,  any  function,  f  (x)-,  is  continuous  in 
the  vicinity  of  a;  =  a,  if,  when  a;  =  a,  f(x)  remains  real  and 
=  fC^^i  ^  finite  constant. 

Theorem  2.  If  ^  =  f(x')  be  a  continuous  function  of  x 
in  the  vicinity  of  a;  =  a,  then  the  inverse  function 

is  a  continuous  function  of  ^  in  the  vicinity  of  the  value 

For  y  =  f  {x)  can  be  represented  by  a  curve  which  is 
continuous  at  (a,  6),  and 

is  represented  by  the  same  curve  in  the  vicinity  of  (a,  6).* 

Cor.     If  f{x)  =  f{a\ 

then  one  value  of  x  approaches  the  limit  a. 

*  A  rigorous  algebraic  proof  of  the  continuity  of  an  inverse  function  will 
be  found  in  the  appendix. 


30  DlFFi:R£NTlAL   CALCULUS  [Cn.  I. 

Theorem  3.  If  two  functions  be  continuous  at  x  =  a, 
then  their  sum,  difference,  and  product  are  continuous  func- 
tions at  X  =  a,  and  also  their  quotient,  provided  the  denomi- 
nator does  not  vanish  at  a;=  a.  Tliis  follows  from  Th.  8,  9, 
Art.  9. 

Cor.  1.  The  product  of  any  finite  number  of  functions, 
each  of  which  is  continuous  at  a:  =  a,  is  continuous  at  x  —  a. 

Cor.  2.  If  ^(x)  be  continuous,  and  <f)(^a)^0, — —  is 
continuous  at  a;  =  a.  r     v 

Theorem  4.  The  algebraic  function  x",  in  which  n  is 
any  commensural)le  number,  is  continuous  for  all  values  of  x 
not  infinite. 

(1)  Let  n  be  a  positive  integer;  then  theorem  3  applies. 

(2)  Let  n  be  the  reciprocal  of  a  positive  integer ;  and  let 

1 

then  x^y'i; 

hence,  by  (1),  a;  is  a  continuous  function  of  ?/,  and  by  theorem 
2,  y  is  a  continuous  function  of  x. 

(3)  Let  n  be  a  positive  fraction,  - ;  then  xf  is  continuous 

by  (2),  and  {x''y  is  continuous  by  (1). 

(4)  Let  n  be  any  negative  commensurable  number ;  then 
corollary  2  of  theorem  3  applies. 

Cor.  a  rational  integral  function  is  finite  and  continuous 
for  all  finite  values  of  the  variable.     (Theorems  3,  4.) 

Lemma.  When  x  approaches  zero  as  a  limit,  then  the 
exponential  function  a^  approaches  unity  as  a  limit : 

i.e.^  if  X  =  Q,  then  a^  =  1. 


H.]  FUNDAMENTAL   PRINCIPLES  31 

For,  let  h  be  any  assigned  positive  number,  and  let  a-  =  -,  in 

n 

which  n  can  become  as  large  as  desired  ;  then  it  is  evidently 
possible  to  choose  n  so  large  that  (1-1-  /f)"  shall  exceed  the 
number  a  (if  n  be  tirst  supposed  positive), 

i.e.,  (l+/0">«, 

and  1  +  h>  a", 

hence  a^  —  1  <  h. 

Thus  the  exponent  x  has  been  chosen  so  small  that  a*  —  1 
is  less  than  the  assigned  number,  i.e..  a'  —  1  =  0,  and  a^  =  1, 
when  x  =  ()  from  the  positive  side.  The  proof  for  the  nega- 
tive approach  follows  from  the  identity  a~^  •  a-^  =  1,  and' 
theorem  9,  p.  16. 

Theorem  5.  The  exponential  function  a'  is  a  continuous 
function  of  a;,  when  x  is  not  infinite,  provided  a  is  positive, 

i.e.,  a'+*-a^  =  0,   when  h  =  0. 

For  a^^'^  -a^=  a-'  (a*  -  1), 

but  a''  —  1  =  0,  when  A  =  0,  by  lemma, 

hence  a^^*  —  a^  =  0,   when  /*  =  0. 

and  a-'  is  a  continuous  function  of  x. 

Thedkem  t).  The  function  \ogaX  is  continuous  when  x 
lies  between  zero  and  positive  infinity  where  a  is  positive. 

For,  let  i/  =  logaX, 

then  '  X  =  ay; 

hence,  by  theorem  5,  a:  is  a  continuous  function  of  y.  when  7/ 
lies  between  —  3o  and  +  x,  that  is  x  between  0  and  -f  x. 
Therefore,  by  theorem  2,  ^  is  a  continuous  function  of  a;, 
when  X  lies  between  0  and  -|-  ». 


32  DIFFERENTIAL   CALCULUS  [Ch.  I. 

Cor.  1.  If  w,  V  be  two  continuous  variables,  then  u""  =  a^ 
when  u  =  a,  where  a  is  positive,  and  v  =  b. 

For  log  u  =  log  a, 

and,  since  v  =  b, 

hence  v\ogu  =  b  log  a, 

that  is,  log  w"  =  log  a*, 

therefore  7^"  =  a*,  when  u  =  a,  v  =  b.  [Th.  2,  Cor. 

Cor.  2.  If  u,  v  be  continuous  functions  of  x,  u"  is  a  con- 
tinuous function  of  a:.      (Th.  1,  Cor.) 

Cor.  3.  If  a:  be  a  continuous  variable,  a:"  is  a  continuous 
function  of  x,  when  n  is  either  commensurable  or  incom- 
mensurable.    This  corollary  is  a  generalization  of  theorem  4. 

Theorem  7.  The  functions  sinx,  cos  a:  are  continuous  for 
all  finite  values  of  x ; 

i.e.,  sin  {x  +  h)  —  sin  x  =  0,  when  h  =  0, 

for  sin  (a:  +  h)  —  siu  a-  =  2  cos  (x  +  ^  h)  sin  |  h, 

but  sin  1^  =  0  when  A  =  0,  and  cos  (a;  +  ^  h)  is  not  infinite, 
hence  sin  (a;  -f  A)  —  sin  a;  =  0  when  A  =  0, 

that  is,  sin  a;  is  continuous. 

Similarly  for  cos  a;. 

EXERCISES 

1.  Prove  that  tan  x,  secx  are  continuous  functions  of  x  for  all  values 
except  X  =  l(2n  +  l)7r,  n  being  any  integer. 

2.  Prove  that  cot  x,  esc  x  are  continuous  functions  of  x  for  all  values 
except  X  =  HTT,  n  being  any  integer. 

3.  Find  the  bounds  of  continuity  of  each  inverse  trigonometric  func- 
tion.    Draw  the  graph,  and  show  the  continuity  of  each  of  the  multiple 

values. 

1 

4.  Show  that  2'  is  not  continuous  at  x  =  0.  Let  x  successively 
approach  zero  from  positive  and  negative  values. 


U-15.]  FUNDAMENTAL  PRINCIPLED  33 

15.  Comparison  of  simultaneous  infinitesimal  increments 
of  two  related  variables.  The  last  few  articles  were  con- 
cerned with  the  principles  to  be  used  in  comparing  any  two 
infinitesimals.  In  the  illustrations  given,  the  law  by  which 
each  variable  approached  zero  was  assigned,  or  else  the  two 
variables  were  connected  by  a  fixed  relation ;  and  the  object 
was  to  find  the  limit  of  their  ratio.  The  value  of  this 
limit  gave  the  relative  importance  of  the  infinitesimals. 

In  the  present  article  the  particular  infinitesimals  com- 
pared are  not  the  principal  variables  (a;,  y)  themselves,  but 
simultaneous  increments  (A,  k}  of  these  variables,  as  they 
start  out  from  given  values  (a^j,  y^)  and  vary  in  an  assigned 
manner;  as  in  the  familiar  instance  of  the  abscissa  and 
ordinate  of  a  given  curve. 

The  variables  x,  y  are  then  to  be  replaced  by  their  equiva- 
lents x^-\-h^  y-^-\-k\  in  which  the  increments  A,  k  are  them- 
selves variables,  and  can,  if  desired,  be  both  made  to  approach 
zero  as  a  limit ;  for  since  y  is  supposed  to  be  a  continuous 
function  of  a:,  its  increment  can  be  made  as  small  as  desired 
by  taking  the  increment  of  x  sufficiently  small. 

The  determination  of  the  limit  of  the  ratio  of  A:  to  A,  as  A 
approaches  zero,  subject  to  an  assigned  relation  betAveen  x 
and  y,  is  the  fundamental  problem  of  the  Differential 
Calculus. 

M.g.^  let  the  relation  be 

y  =  ^\ 

let  STj,  yj  be  simultaneous  values  of  the  variables  x,  y ;  and 
when  X  changes  to  the  value  x-^  +  A,  let  y  change  to  the 
value  y^-\-k\  then 

yj  -f  ^  =  (a:^  +  A)2  =  a^i^  +  2  x^  +  A^; 
hence  A:  =  2  x^  4-  A^. 


34  DIFFEUEM'IAL   CALCULUS  [Cii.  I. 

This  is  a  relation  connecting  the  incieuients  h,  k. 

Here  it  is  to  be  observed  that  the  relation  between  the 
infinitesimals  /i,  k  is  not  directly  given,  but  has  first  to  be 
derived  from  the  known  relation  between  x  and  y. 

Let  it  next  be  required  to  compare  these  simultaneous 
increments  by  finding  the  limit  of  their  ratio  when  they 
approach  the  limit  zero. 

By  division, 

hence,  by  Art.  9,  theorem  5, 

liin   /'  _  9 

This  result  may  be  expressed  in  familiar  language  by 
saying  that  when  x  increases  through  the  value  r^,  then  y 
increases  2x^  times  as  much  as  x-,  and  thus  when  x  continues 
to  increase  uniformly,  y  increases  more  and  more  rapidly. 
For  instance,  when  x  passes  through  the  value  4,  and  y 
through  the  value  16,  the  limit  of  the  ratio  of  their  incre- 
ments is  8,  and  hence  y  is  changing  8  times  as  fast  as  x ;  but 
when  x  is  passing  through  5,  and  y  through  25,  the  limit  of 
the  ratio  of  their  increments  is  10,  and  y  is  changing  10 
times  as  fast  as  x. 

The  following  table  will  numerically  illustrate  the  fact 
that  the  ratio  of  the  infinitesimal  increments  A,  k  approaches 
nearer  and  nearer  to  some  definite  limit  when  h  and  k  both 
approach  the  limit  zero. 

Let  ajj,  the  initial  value  of  a;,  be  4;  then  ^j,  the  initial 
value  of  y,  is  16.  Let  A,  the  increment  of  x,  be  1 ;  then  k, 
the  corresponding  increm.ent  of  y,  is  found  from 

164-A:  =  (4  +  1)2; 


15.] 


FUNDAMENTAL  PRINCIPLES 


35 


thus  k=9,  and  y  =  9-     Next  let  h  be  successively  diminished 
to  the  values  .8,  .6,  .4,  •••;  then  the  corresponding  values  of 

k  and  of  -  are  as  shown  in  the  table : 
h 


X  =  4  +  A 

y  =  16  +  k 

k 

k 
h 

4+   1 

25 

9 

9 

4 +  .8 

23.04 

7.04 

8.8 

4 +  .6 

21.16 

5.16 

8.6 

4 +  .4 

19.36 

3.36 

8.4 

4 +  .2 

17.64 

1.64 

8.2 

4  +  .1 

16.81 

.81 

8.1 

4 +  .01 

16.0801 

.0801 

8.01 

4  +  A 

16  +  8  A  +  A2 

8A  +  /*2 

8-i- A 

Thus  the  ratio  of  corresponding  increments  takes  the 
successive  values  8.8,  8.6,  8.4,  8.2,  8.1,  8.01,  ...,  and  can  be 
brought  as  near  to  8  as  desired  by  taking  h  small  enough. 

As  another  example  let  the  relation  between  x  and  y  be 


then 


y'^  =  x<*, 


y^-  =  x,« 

(^1  +  ky  =  (Xj  +  A)8, 

hence,  by  expansion  and  subtraction, 

2  y^k  +  k^  =  li  x^Vi  +  3  x^h^  +  A», 

k(2y^  +  k)  =  h  (3  Xi2  +  3  x,A  +  A2), 

k_3x^^  +  3x^h  +  h^ 
h~  2yi  +  k 


Therefore 


hm  Y  =  lira  — i-^5 ^ ,  ask  =  O.K  —  0, 


and,  by  Art.  10,  theorem  10, 

,.     k      3  r « 
h      2.^1 


1>IFF.  CALC. 4 


36 


DIFFERENTIAL   CALCULUS 


[Ch.  1. 


The  "  initial  values  "  of  x,  y  have  been  written  with  sub- 
scripts to  show  that  only  the  increments  (A,  k)  vary  during 
the  algebraic  process,  and  also  to  emphasize  the  fact  that  the 
limit  of  the  ratio  of  the  simultaneous  increments  depends  on 
the  particular  values  through  which  the  variables  are  pass- 
ing, when  they  are  supposed  to  take  these  increments. 
With  this  understanding  the  subscripts  will  hereafter  be 
omitted.  Moreover,  the  increments  A,  k  will,  for  greater 
distinctness,  be  denoted  by  the  symbols  Aa;,  Ay,  read  "incre- 
ment of  a:,"  "increment  of  y."  The  symbol  A  is  derived 
from  the  initial  letter  of  the  word  difference^  as  the  increment 
of  a  variable,  in  passing  from  one  value  to  another,  is  ob- 
tained by  subtracting  the  first  value  from  the  second. 

A?/ 


Ex.  1.   If  x2  +  2/2  =  a\  find  lim 


Ax' 


Let  the  initial  values  of  the  vari- 


hence 
and 


ables  be  denoted  by  x,  y,  and  let  the  variables  take  the  respective  incre- 
ments Ax,  Ay,  so  that  their  new  values  x  +  Ax,  //  +  Ay  shall  still  satisfy 
the  given  relation,  then 

(x  +  Ax)2  +  (y  +  Ay)2  =  a\ 
By  expansion,  and  subtraction, 

2  X  •  Ax  +  (Ax)2  +  2  y  .  Ay  +  (A.v)2  =  0, 
Ax  (2  X  +  Ax)  =  —  Ay  (2  y  +  Ay), 
A^  _  _  2  X  +  Ax_ 
Ax  2  y  +  Ay 

lim      Ay  _  _     lim     2  x  +  Ax  _  _  i 
Ax  =  0  ^  ~      Ax  =  0  2  y  +  Ay  ~      y 

The  negative  sign  indicates  that  when 
Ax,  and  the  ratio  x :  y,  are  positive.  Ay  is 
negative,  that  is,  an  increase  in  x  produces 
a  decrease  in  y.  This  may  be  illustrated 
geometrically  by  drawing  the  circle  whose 
^  equation  is  ^2  +  y2  _  ^2  (Fig.  5). 


I'herefore 


Fia.  5. 


Ex.  2.    If  x2  +  y  =  y2  -  2  X, 

lim     Ay  _  2  x  +  2, 
Aa;  =  0Ax~2y-l 


prove 


16-16.]  FUNDAMENTAL  PRINCIPLES  37 

Similarly  when  the  relation  between  x  and  y  is  given  in 
the  explicit  functional  form 

y  =  <f)  (x\ 

then  1/  +  Ai/  =  <^  (x  +  Aa;), 

and  Ay  =  ^  (a;  +  Aa;)  —  <f)  (x}  =  A  ^  (a;), 

hence  lim  ^  =  lun  H^  +  ^)  -  H^). 

Aa;  Aa; 

When  the  form  of  0  is  given,  the  limit  of  this  ratio  can 
be  evaluated,  and  expressed  as  a  function  of  x;  and  this 
function  is  then  called  the  derivative  of  the  function  (f>(x') 
with  regard  to  the  independent  variable  x. 

The  formal  definition  of  the  derivative  of  a  function  with 
regard  to  its  variable  is  given  in  the  next  article. 

16.  Definition  of  a  derivative. 

If  to  a  variable  a  small  increment  be  given,  and  if  the 
corresponding  increment  of  a  continuous  function  of  the 
variable  be  determined,  then  the  limit  of  the  ratio  of  the  in- 
crement of  the  function  to  the  increment  of  the  variable, 
when  the  latter  increment  approaches  the  limit  zero,  is  called 
the  derivative  of  the  function  as  to  the  variable. 

Let  <l)(x}  be  a  finite  and  continuous  function  of  x,  and  Aa; 
a  small  increment  given  to  a;,  then  the  derivative  of  <f)  (x)  as 
to  X  is 

lim     \4>{x  +  Aa;)  -  4>(x^\  _     Um     A<^(a;) 
Ax  =  0|  ^x  )—  Aa:=0       /^^ 

It   is    important    to    distinguish    between    lim   -~^ and 

— ^. —  \,       1  that  is,  between  the  limit  of  the  ratio  of  two 
lim  Aa; 

infinitesimals,  and  the  ratio  of  their  limits.     The  latter  is 
indeterminate  of  the  form  -  and  may  have  any  value ;  but 


38  DIFFERENTIAL   CALCULUS  [Ch.  I. 

the  former  has  usually  a  determinate  value,  as  illustrated  in 
the  examples  of  the  last  article. 

EXERCISES 

1,  Find  the  derivative  of  x'^  —  2  x  as  to  x. 

2,  Find  the  derivative  of  3  j;"^  —  4  x  +  3  as  to  x. 

3.  Find  the  derivative  of  —  as  to  x. 

4  X 

4.  Find  the  derivative  oi  x*  —  2  +  ^  as  to  x. 

x^ 

17.  Geometrical  illustrations  of  a  derivative. 

Some  conception  of  the  meaning  and  use  of  a  derivative 
will  be  afforded  by  one  or  two  geometrical  illustrations. 

Let  y  =  0  (x)  be  a  function  of  x  that  remains  finite  and ' 
continuous  for  all  values  of  x  between  certain  assigned  con- 
stants a  and  h ;  and  let  the  variables  x,  y  be  taken  as 
the  rectangular  coordinates  of  a  moving  point ;  then  the 
relation  between  x  and  y  is  represented  graphically,  within 
the  assigned  bounds  of  continuity  by  the  curve  whose  equa- 
tion is 

y  =  ^{x). 

Let  (zj,  yj),  (ajg,  y^  be  the  coordinates  of  two  points 
Pj,  Pg  o'^  ^his  curve;  then  it  is  evident  that  the  ratio 

'^'^  ~  y^  =  tan  a, 

wherein  a  is  the  inclination  angle  of  the  secant  line  P^P^y 
to  the  2;-axis.  Let  P^  be  moved  nearer  and  nearer  to  coinci- 
dence with  Pj,  so  that  x^  =  x^^  ^2  =  ^1'  then  the  secant  line 
P^Pi  approaches  nearer  and  nearer  to  coincidence  with  the 
tangent  line  drawn  at  the  point  Pj,  and  the  inclination  angle 


16-17.] 


FUNDAMENTAL  PRINCIPLES 


39 


Fio.  6. 


P2(x„y,) 


(«)  of  the  secant  approaches  as  a  limit  the  inclination  angle 
(</>)  of  the  tangent  line. 
Hence,    by    theorem    7^ 
and  Ex.  1,  Art.  14, 

tan  a  =  tan  <f>. 
Thus    ^^^  =  tan<^, 

when  x^  =x^,  y^  =  y^ 

This  can  also  be  seen 
from  the  similar  triangles 

KSP^  and  P^MP^. 

The  proportion 

P^M^  SP, 

MP.     KS 


is  true,  whatever  be  the  position  of  P^.  When  P^  ap- 
proaches to  coincidence  with   Pj,   P^M=  0,  MP^  =  0,  but 

SP 

their  ratio  approaches  -^^,  which  is  tan  <^. 

TS 

It  may  be  observed  that  if  x^  be  put  directly  equal  to 
Xj,  and  2/2  to  yj,  the  ratio  on  the  left  would,  in  general, 

assume  the  indeterminate  form  -,  as  in  other  cases  of  find- 
ing the  limit  of  the  ratio  of  two  infinitesimals ;  but  it  has 
just  been  shown  that  the  ratio  of  the  infinitesimals  y^  —  y^, 
a-2  —  x^  has,  nevertheless,  a  determinate  limit  measured  by 
tan  0. 

They  are    thus  infinitesimals   of   the  same  order  except 
when  <^  is  0  or  — • 

If  the  differences  x^  —  x^,  y<^  —  y^  be  denoted  by  Aa:,  Ay, 
then  x^:=x^  +  Aa;,     y^  =  y^-\-  ^.y\ 


40 


l)lft't:Rt]NTtAL   CALGULVrd 


[ch.  i 


but,  since  ^  =  <j>  (^)' 

hence   the   ratio   of    the   simultaneous   increments   may   be 
written  in  the  various  forms 

^1  "^2        "^l  ^ 


In  the  last  form,  x  is  regarded  as  the  independent  variable, 
and  Aa;  its  independent  increment ;  and  the  numerator  is 
the  increment  of  the  function  ^(a;),  caused  by  the  change 
of  X  from  the  value  x^  to  the  value  x^  +  Aa;.  The  limit 
of  this  ratio,  as  Aa;  =  0,  is  the  value  of  the  derivative  of 
the  function  (f}(x),  when  x  has  the  value  x^  Here  x^  stands 
for  any  assigned  value  of  x.  Thus  the  derivative  of  any 
continuous  function  <f>  (x)  is  another  function  of  x  which 
measures  the  slope  of  the  tangent  to  the  curve  y  =  ^  (x), 

drawn  at  the  point  whose  abscissa  is  x. 

2 
PjX.    Find  the  slope  of  the  tangent  line  to  the  curve    ?/  =  —  at  the 

point  (1,  2). 


Here 


.        ■         lim      (x  +  \xy 
tan  ©  =  A      .  n  ^^ — ■ ^— 


_    lim      -  2  (2  X  4-  Ax)  ^      4 
Ax  =  0    a.-2(x  +  Ax)2  x8 

Hence  tan  ^  =  —  4,  when  x  =  1 ;  and  the  equation  of  the  tangent  line 
at  the  point  (1,  2)  is  y-2^-i(x-  1).  [Cf.  A.G.,  Art.  53. 

^  As  another  illustration,  let  the 

coordinates  of  P  be  (x,  ^),  and 
those  of  Q,(x  4-  Ax,  y  +  A^/)  ;  tlien 
MN=  PR  =  Ax,  and  P>S=  RQ 
=  Ay.  Let  the  area  OAPM  be 
denoted  by  z,  then  z  is  evidently 
some  function  of  the  abscissa  a;; 
also  let  area  OAQN,  =  2  +  Az,  theu  area  MNQP  =  Az,  is  the 


17  18.]  FUNDAMENTAL   PRINCIPLES  41 

increment  taken  by  the  function  z,  when  x  takes  the  incre- 
ment Ax;  but  MNPQ  lies  between  the  rectangles  MR^  MQ^ 

hence  i/ Ax  <  Az  <  (i/ +  Ai/)  Ax, 

and  1/  <-~ <y  +  Ay. 

Therefore,  when  Ax,  A^,  Az  all  =  0, 

v     Az 
lim  —  =  y. 

Ax     ^ 

Thus  if  the  ordinate  and  the  area  be  each  expressed  as 
functions  of  the  abscissa,  the  derivative  of  the  area  furtction 
with  regard  to  the  abscissa  is  equal  to  the  ordinate  function. 

Ex.  If  the  area  included  between  a  curve,  the  axis  of  y,  and  the 
ordinate  whose  abscissa  is  a:,  be  given  by  the  equation 

z  =  dfi, 

find  the  equation  of  the  curve. 

Here  y  =  lim  ^  =  .  ""^  „  (x  +  Ax)8  -  x^ 

"  Ax      ^^  =  0  Ar 

=  ^^1  0  [3  x^  +  3  xAx  +  (Ax)2]  =  3  x\ 

.  18.  The  operation  of  differentiation.  It  has  been  seen  in  a 
number  of  examples  that  when,  on  a  given  function  <f>{x), 
the  operation  indicated  by 

lim     <t>(x-\-  Ax)  —  <f> (a;) 
Ax  =  0  ^ 

is  performed,  the  result  of  the  operation  is  another  function 
of  X.  This  function  may  have  properties  similar  to  those  of 
<f>(x)-,  or  it  may  be  of  an  entirely  different  class. 

The  above  indicated  operation  is  for  brevity  denoted  by 


42  DIFFERENTIAL  CALCULUS                      [Ch.  I. 

the  symbol    ^  ,  and  the  resulting  derivative  function  by 
<f>'(x) ;  thus 

<^4*  (^)  _    lira  ^</>  (^)  _    lim     </>  (2:  +  Aa;)  —  <^  (a;) 


^3,      -Az  =  0      ^^      -Ax  =  0  ^^ 


<^'(^)- 


The  process   of    performing   this   indicated   operation   is 
called  the  differentiation  of  (f>{x^  with  regard  to  x.     The 

symbol  *  —-,  when  spoken  of  separately,  is  called  the  differ- 
entiating operator,  and  expresses  that  any  function  written 
after  the  d  is  to  be  differentiated  witli  regard  to  x,  just  as 
the  symbol  cos  prefixed  to  ^(a:)  indicates  that  the  latter  is 
to  have  a  certain  operation  performed  upon  it ;  namely,  that 
of  finding  its  cosine. 

The  process  of  differentiating  (j>{x)  consists  of  the  follow- 
ing steps : 

1.  Obtain  ^(x  +  Ax)  by  changing  x  into  x  +  Ax  in  ^(x). 

2.  B^ind  A(f)(x^  by  subtracting  </)(a?)  from  (f>(x  +  Ax}. 

3.  Divide  this  difference  A(f>{x)  by  Ax. 

4.  Find  the  limit  of  the  quotient     ^        when  Ax  =  0. 

This  series   of  steps   should  be   memorized.     In  words, 
these  four  steps  can  be  expressed  as  follows : 

1.  Give  a  small  increment  to  the  variable. 

2.  Compute  the  resulting  increment  of  the  function. 

3.  Divide  the  increment  of  the  function  by  the  incre- 
ment of  the  variable. 

4.  Obtain  the  limit  of  this  quotient  as  the  increment  of 
the  variable  approaches  zero. 


♦  This  symbol  is  sometimes  replaced  by  the  jingle  letter  D. 


18-19.]  FUNDAMENTAL  PRINCIPLES  43 

EXERCISES 
Find  the  derivatives  of  the  foUowing  functions : 

1.  5  ?/3  _  2  y  +  6  as  to  y ;  3.   8  u^  _  4  «  +  10  as  to  2  « ; 

2.  7  <2  -  4  <  -  11  <8  as  to  < ;  4.   2  x^  -  5  a;  +  6  as  to  x  -  3. 

This  process  will  be  applied  in  the  next  chapter  to  all  the 
classes  of  functions  whose  continuity  within  certain  inter- 
vals has  been  established  in  Art.  14;  and  it  will  be  found 
that  for  each  of  them  a  derivative  function  exists;  that  is, 

that  lim      ^        has  a  determinate  and  unique  value,  and 

that  the  curve  y  =  <f>  (x^  has  a  definite  tangent  within  the 
range  of  continuity  of  the  function. 

A  few  curious  functions  have  been  devised,  which  are  continuous  and 
yet  possess  no  definite  derivative ;  but  they  do  not  present  themselves  in 
any  of  the  ordinary  uses  of  the  Calculus.    Again,  there  are  a  few  functions 

for  which  lim     ^      has  a  certain  value  when  Ax  =  0  from  the  positive 

side,  and  a  different  value  when  A.r  =  0  from  the  negative  side ;  the  de- 
rivative is  then  said  to  be  non-unifjue.     [Cf.  Ex.  11,  p.  282.] 

Functions  that  possess  a  unique  derivative  within  an  as- 
signed interval  are  said  to  be  differentiahle  in  that  interval. 

Ex.  Show  that  a  function  is  not  differentiahle  at  a  discontinuity 
(Art.  6). 

19.  Increasing  and  decreasing  functions.  A  good  example 
of  the  use  of  the  derivative  is  its  application  to  finding  the 
intervals  of  increasing  or  decreasing  for  a  given  function. 

A  function  is  called  an  increasing  function  if  it  increases 
as  the  variable  increases  and  decreases  as  the  variable  de- 
creases. A  function  is  called  a  decreasing  function  if  it  de- 
creases as  the  variable  increases,  and  increases  as  the  variable 
decreases. 

E.g.,  the  function  x^  +  4  decreases  as  x  increases  from  —  00  to  0,  but 
it  increases  as  x  increases  from  0  to  +  qo.     Thus  x^  +  4  is  a  decreasing 


44 


DIFFERENTIAL   CALCULUS 


[Cii.  I. 


function  while  x  is  negative,  and  an  increasing  function  while  x  is  posi- 
tive.   This  is  well  shown  by  the  locus  of  the  equation  >/  —  x^  +  4(Fig.  8). 

F 


Fig.  8. 


Fig.  !). 


Again,  the  form  of  the  curve  y  =  -  shows  that  -  is  a  decreasing  func- 

X  X 

tion,  as  x  passes  from  —  oo  to  0,  and  als  >  a  decreasing  function,  as  x 
passes  from  0  to  +  go.  When  x  passes  through  0,  the  function  changes 
discontinuously  from  the  value  —  oo  to  the  value  +  co  (Fig.  9). 

Most  functions  are 
increasing  functions 
for  some  values  of  the 
variable,  and  decreas- 
ing functions  for 
others. 


E.g.,    y/'2  rx  —  x^   is    an 
'"■     ■  increasing    function    from 

X  =  0  to  a;  =  r,  and  a  decreasing  function  from  x  =  rtox  =  2r  (Fig.  10). 

A  function  is  said  to  be  an  increasing  function  in  the 
neighborhood  of  a  given  value  of  x  if  it  increases  as  x 
increases  through  a  small  interval  including  this  value ; 
similarly  for  a  decreasing  function. 


19-20.] 


FUNDAMENTAL  PBtNClPLES 


45 


20.  Algebraic  test  of  the  intervals  of  increasing  and  de- 
creasing. Let  y  =  <f){x)  be  a  function  of  x,  and  let  it  be  real, 
continuous  and  differentiable  for  all  values  of  x  from  a  to  b; 
then  by  definition  i/  is  increasing  or  decreasing  at  a  point 
X  =  a?!,  according  as 

(f){x^  +  Aa;)  —  (f>{x^) 

is  positive  or  negative,  where  Ax  is  a  small  positive  number. 
The  sign  of  this  expression  is  not  changed  if  it  be  divided 
by  Ax,  no  matter  how  small  Ax  may  be ;  hence  <f>{x}  is  an 
increasing  or  a  decreasing  function  at  the  value  iCj,  accord- 
ing as 

di/ _    Urn    i  <^(a^i  +  Ax)-  (f>(x^)  \  _  ^,^^  ^ 


•V 


Ax 


is  positive  or  negative. 

Thus  the  intervals  in  which  ^(a;)  is  an  increasing  function 
are  the  same  as  the  intervals  in  which  (f>'(x}  is  positive. 

E.g.,  to  find  the  intervals  in  which  the  function 
<^(x)  =  2  x8  -  9  a:2  +  12  X  -  6 
is  increasing  or  decreasing.     The  derivative  is 

<f>'(x)  =  6z2  -  18x  +  12  =  6(x  -  l)(x  -  2); 
hence,  as  x  passes  from  —  oo  to  1,  the  derived  function  <t*'(^),  is  positive 
and  <^(x)  increases  from  ^(—  oo)  to  <^(0» 
i.e.,  from  <f>  =  —xto<f>  =  —l;  as  x  passes 
from  1  to  2,  <t>'{x)  is  negative,  and  <^(x) 
decreases  from  <^(1)  to  ^(2);  i.e.,  from 
—  1  to  —  2 ;  and  as  x  passes  from  2  to  +  oo, 
<^'(x)  is  positive,  and  ^(x)  increa.ses  from 
«^(2)  to  <^(x) ;  i.e.,  from  —  2  to  +  oo. 
The  locus  of  the  equation  ij  =  ff>(x)  is 
shown  in  figure  11.  At  points  where 
<^'(x)  =  0,  the  function  <f>(x)  is  neither 
increasing  nor  decreasing.  At  such  points 
the  tangent  is  parallel  to  the  axis  of  x. 
Thus  in  this  illustration,  at  x  =  1,  x  =  2, 
the  tangent  is  parallel  to  the  x-axis.  Fio.  ii. 


46  DIFFERENTIAL   CALCULUS  [Cii.  I. 

EXERCISES 

1.  Find  the  intervals  of  increasing  and  decreasing  for  the  function 

Here  <i>'{x)  =  3a:2  +  4x  +  1  =  (:3t  +  l)(x  +  1). 

The  function  increases  from  x=:  —  cotoa:  =  —  1;  decreases  from  a;  =  —  1 

to  X  =  —  I ;  increases  from  x  =  —  |  to  x  =  cc. 

2.  Find  the  intervals  of  increasing  and  decreasing  for  the  function 

y  =  x^  —  2  x^  +  X  —  4, 
and  show  where  the  curve  is  parallel  to  the  x-axis. 

3.  At  how  many  points  can  the  slope  of  the  tangent  to  the  curve 

y  =  2x8-3x2+l 
be  1?    -  1  ?     Find  the  points. 

4.  Compute  the  angle  at  which  the  following  curves  intersect : 

y  =  3  x2  -  1,      y  =  2  x2  +  3.  [Cf.  A.G.,  p.  164. 

21.  Differentiation  of  a  function  of  a  function.  Suppose 
that  y,  instead  of  being  given  directly  as  a  function  of  a:, 
is  expressed  as  a  function  of  another  variable  w,  which  is 
itself  expressed  as  a  function  of  x ;  and  let  it  be  required  to 
find  the  derivative  oi  y  with  regard  to  the  independent 
variable  x. 

X/ct  y  =f(u)^  in  which  w  is  a  function  of  x.  Suppose 
that  X  passes  through  an  assigned  value  x^ ;  and  let  u  pass 
through  a  corresponding  value  w^ ;  and  y^  ia  consequence, 
through  a  value  y^  When  x  changes  to  the  valuQ  x^  +  Ar, 
let  u  and  y,  under  the  given  relations,  change  continuously 
to  the  values  u^  +  Aw,  y^  +  Ay ;  then 

Ay  _  Ay   Aw  _  f(u  +  A«)  —f(u)    Au^ 
Ax ~  Au   Ax~  Aw  Aa;' 

hence,  equating  limits, 

dy  _  dy   du  _  df(u)    du 
dx     du   dx        du      dx 


20-22]  FUNDAMENTAL  PRINCIPLES  47 

in  which  the  combination  of  values  (x  =  a:j,  w  =  Mj,  y  =  y^) 
is  to  be  substituted. 

The  derivative  of  a  function  of  u  with  regard  to  x  is  equal 
to  the  product  of  the  derivative  of  the  function  with  regard  to 
w,  and  the  derivative  of  u  with  regard  to  x;  each  derivative 
being  estimated  at  the  same  combination  of  corresponding 
values  of  the  three  variables. 


The  given  functions  may  be  multiple-valued,  such  as  y  =  V«2  —  u% 
u  =  sin-*j:.     Then  when  any  assigned  value  x^  is  given  to  x,  the  functions 

«  and  —  take  multiple  values ;  let  one  of  the  branches  of  u  be  specified ; 

dx 
and  let  u^  be  the  value  of  «  on  this  branch,  corresponding  to  x  =  Xy 

When  the  value  u^  is  given  to  u,  the  functions  y  and  -^  take  multiple 

values;  let  the  value  of  y  on  a  specified  branch  be  y^     Then,  by  the 

theorem,  one  of  the  values  of  —  taken  at  x  =  Xj,  multiplied  by  one  of 

the  values  of  -^  taken  at  (x  =  x,,  m  =  w,),  will  give  one  of  the  values  of 
,  <iu 

-^  taken  at  (x  =  Xj,  y  =  ^j),  and  these  are  the  respective  unique  values 

of  the  three  derivatives  taken  at  the  specified  combination  (x  =  Xj,  «  =  Uj, 
y  =  yi).  This  combination  is  represented  geometrically  in  three  dimen- 
sions by  one  of  the  points  of  intersection  of  the  plane  x  =  Xj  with  the 
intersection-curve  of  the  two  surfaces  that  represent  the  given  functions. 

Ex.  1.   Given       y  =  3  u^  -  1,  «  =  3x2  -i-  4;    find  ^. 

dy  du 

-f-z=Qu,   -J-  =  6x; 
dti  ax 

^  =  f.^  =  S6ux  =  S6x(Sx^  +  i). 
ax      ilu    ax 

Ex.  2.   Given       y  =  ^u'i  -  iu  +  5,  u  =  2x^  -  5;  find  $^- 

ax 

22.   Differentiation  of  inverse  functions.     Relation  between 

du  dx 

-^  and  -J-.     When  y  =  /(a;)  is  a  continuous  and  differen- 

tiable  function  of  a;,  the  symbol  -^  stands  for  the  numerical 


48  DIFFERENTIAL   CALCULUS  [Ch.  I 

measure  of  the  limit  of  the  ratio  of  an  increment  of  y  to  an 
assigned  increment  of  x.  Next,  let  y  be  taken  as  the  inde- 
pendent variable  ;  then  the  inverse  function  x  =  f~^iy}  is 
a  continuous  function  of  y ;  and  if  a  small  increment  be 
given  to  y,  it  is  required  to  find  the  limit  of  the  ratio  of  the 
resulting  increment  of  x  to  the  assigned  increment  of  y. 

Let  x,  y  have  the  initial  values  x^,  y^,  and  let  the  variables 
change,  subject  to  the  given  relation,  so  as  to  assume  the 
values 

x^  +  Ax,yj^  +  A«/; 

,,  ,  Ay     Aa;       ^ 

then,  since  -r^  •  -:—  =  1, 

Ax     Ay 

hence,  by  the  theory  of  limits  (Art.  9,  Th.  8,  Cor.), 
dx  _  1 

dx 

in  which  the  two  corresponding  values,  a;  =  a:^,  y  =  y^,  are 
understood  to  be  substituted. 

Thus  ii  y  =  f(x)  be  a  differentiable  function  of  x,  the 
inverse  function  x  =  f~^{y)  is  a  differentiable  function  of 
y,  and  the  derivative  of  x  with  regard  to  y  is  the  reciprocal 
of  the  derivative  of  y  with  regard  to  x,  each  derivative  being 
estimated  at  the  same  pair  of  corresponding  values  of  x 
and  y. 

Note.  Either  variable  may  be  a  multiple-valued  function 
of  tlie  other,  as  in  the  familiar  relation,  a^  +  y^  =  a^. 

When  any  value  Xj  is  given  to  x,  the  functions  y  and  -r- 

take  multiple  values  ;  and,  when  the  corresponding  value  y^ 

is  given  to  y,  the  functions  x  and  — -  take  multiple  values. 

dy 


m 

22.]  FUNDAMENTAL   PRINCIPLES  49 

One  pair  of  values  of  -~  and  of  -^  will  be  reciprocal,  and 
clx  cly  • 

these  will  be  their  respective  values  for  the  combination 

(x  =  x^.  y  =  yi). 

In  geometrical  language,  they  will  belong  to  the  same  point 
(xj,  y^)  of  the  representative  curve. 

Ex.    From  Ex.  1,  p.  36,  find  the  values  of  -^,  —  at  the  four  points 

dx   dy 
(±3,  ±4)  on  the  circle  z^  +  y*  =  25 ;  and  write  down  the  equations  of  the 
four  tangents. 

MISCELLANEOUS  EXERCISES 

1.  Find  lira  ^I^-lil(^±|lasa:  =  ±Qo;  ±1;  ±2;  0. 

2.  If  n  =  GO ,  show  whether  the  theorems  of  limits  apply  to : 

-  -\ 1-  •■•  (to  n  terms)  =  a ; 

1        1 

a«  X  a"^  y.  •••  (to  n  factors)  =  a; 

i        -  1 

an»  X  o"^  X  •••  (to  n  factors)  =0"; 

111  JL  n+l 

an'  X  a"*  X  an^  x  •••  y.  an*  =  a  zn  " 

3.  Draw  graphs  of  a*,  log  x,  log  (x'^  —  x),  tan  x.    Show  discontinuities. 

I         1 

4.  What  kinds  of  discontinuity  have  a",  sin  -,  at  a;  =  0  ? 

5.  What  locus  has  its  area  proportional  to  the  square  of  the  abscissa? 

6.  Show  that  the  perimeter  of  an  inscribed  regular  n-gon  equals 


=  J,  TTT,  as  n  =  00. 


r  •  T1 

sm  — 

2  7irsin      =2  TIT     

n  IT 

L       H         ) 

7.   Prove  that  the  derivative  of  a  constant  is  zero. 


H.T.C 


oxv^. 


CHAPTER   II 

DIFFERENTIATION    OF   THE    ELEMENTARY   FORMS 

23.  In   recent   articles,   the   meaning   of  the   symbol  -^ 

dx 

was  explained  and  illustrated ;  and  a  method  of  expressing 
its  value,  as  a  function  of  a;,  was  exemplified,  in  cases  in 
which  y  was  a  simple  algebraic  function  of  x^  by  direct  use 
of  the  definition.  This  method  is  not  always  the  most 
convenient  one  in  the  differentiation  of  more  complicated 
functions. 

The  present  chapter  will  be  devoted  to  the  establishment 
of  some  general  rules  of  differentiation  which  will,  in  many 
cases,  save  the  trouble  of  going  back  to  the  definition. 

The  next  five  articles  treat  of  the  differentiation  of  alge- 
braic functions  and  of  algebraic  combinations  of  other 
differentiable  functions. 

24.  Differentiation  of  the  product  of  a  constant  and  a  vari- 
able. 


Let 

y  =  cx'. 

then 

y  +  £^y  =  c{x 

-f  Aaj), 

Ay  =  c  (a; 

+  Aa;)- 

■  ex  —  cAx, 

Aa;-''' 

therefore 

1^  =  .. 
dx 

[Art.  9,  Th.  9. 

60 


Ch.  11.23-25.]  DIFFERENTIATION  OF  ELEMENTARY  FORMS    51 

Cor.  1.     \i  y  =  cu,  where  m  is  a  function  of  a;,  then,  by 
Art.  21, 

d(cu)  ^  ^du^  (1) 

dx  dx 

The  derivative  of  the  product  of  a  constant  and  a  variable  is 
equal  to  the  constant  multiplied  by  the  derivative  of  the  variable. 

Cor.  2.     The  operator  -—  and  the  constant  multiplier  c  are 
dx 

commutative  operators. 

Is  this  true  of  the  operators  A  and  c  ? 

25.   Differentiation  of  a  sum. 

Let  y=/(a;)+<^(2:)  +  V^(a;), 

then     y  -h^y  =fix  +  Ax)  +  ^(x  +  Ax)  +  yjr(x  +  Ax)y 

Ay  ^f(x  +  Ax)-f(x)      (l>(x-[-Ax)-<t>(x) 
Ax  Ax  Ax 

^fr(x-]-  Ax)  —  yfr  (x) 
Ax 

therefore,  by  equating  the  limits  of  both  members, 

^  =f'(x)  +  <^'(a:)  +  fix).  [Art.  9,  Th.  7. 

dx 

Cor.  1 .     li  y  =  u  -\-  V  +  w,  in  which  u,  v,  w^  are  functions 

of  X,  then 

A^(iu  +  v  +  w)=^  +  ^  +  ^^  (2^ 

dx  dx     dx     dx 

The  derivative  of  the  sum  of  a  finite  number  of  functions  is 
equal  to  the  sum  of  their  derivatives. 

Cor.  2.     The  operator  — -  is  distributive  as  to  addition. 
dx 

Is  this  true  of  the  operator  A  ? 

mPF.  CALC.  — 5 


52  DIFFERENTIAL   CALCULUS  [Ch.  II. 

If  the  number  of  functions  be  infinite,  theorem  7  of  Art.  9  may  not 
apply,  that  is,  the  limit  of  the  sum  may  not  be  equal  to  the  sum  of  tlie 
limits ;  and  hence  the  derivative  of  the  sum  may  not  be  equal  to  the  sum 
of  the  derivatives.  Thus  the  derivative  of  an  infinite  series  cannot 
always  be  found  by  differentiating  it  term  by  term.  (See  note,  p.  14, 
and  footnote  to  Art.  56.) 

26.  Differentiation  of  a  product. 

Let  1/  =f(x)  </>  (a;), 

then         •  ^  =  f(x  +  Ax)<f>(x  +  Ax)-f(x)<f>{x) 

Ax  Ax 

By  subtracting  and  adding  f(jx)  (f)(^x-\-  Ax}  in  the  numer- 
ator, this  result  may  be  re-arranged  thus : 

Equating  limits,  as  Ax  =  0,  using  Art.  9,  theorems  7,  8, 
and  noting  that  the  first  factor  <f){x  +  Ax)  =  (p  (x)  since 
<l>(x)  is  by  hypothesis  continuous  (Art.  14),  it  follows  that 

Cor.  1.  By  writing  u  =  <^(x)^  v  =f(x),  this  result  can 
be  more  concisely  written, 

dx  dx       dx 

The  derivative  of  the  product  of  two  functions  is  equal  to  the 
sum  of  the  products  of  the  first  factor  by  the  derivative  of  the 
second,  and  the  second  factor  by  the  derivative  of  the  first. 

This  rule  for  differentiating  a  product  of  two  functions 
may  be  stated  thus  :  Differentiate  the  product,  regarding  the 
first  factor  as  constant,  then  regarding  the  second  factor  as 
constant,  and  add  the  two  results. 


25-27.]    DIFFERENTIATION   OF  ELEMENTARY  FORMH  53 

Since  -^(uv)  ^  ?'-r-v,  the  operator  -—  is  not  commutative 
ax  ax  ax 

with  a  variable  multiplier. 

Cor.  2.     To  find  the  derivative  of  the  product  of  three 

functions 

i/  =  <f>ix)0ix)fix). 

Let  /(a-)=^(a;)i|r(a:),  ' 

then  i/  =  (f)  {x)f(x), 

hence  ^^-^^^^  «^'(a:)+  ,^(2:)/'(2;), 

but  /  (x)  =  e  (x)  f  (x)  +  yjr  (x)  0'  {x)  ; 

hence,  substituting  the  values  for  f(x)^f'{x)^ 

^  =  y^(x^  (f,(x)  e'(x)  +  ^(a;)  <j>(x)  ^'(a:)  +  eix^y^rix)  <l>'{x)  ; 
ax 

and  so  on,  for  any  finite  number  of  factors. 
This  result  can  also  be  written  in  the  form 

dL^A*^  =  t,r  "f^  +  vw  *^  +  um^.  (4) 

dx  dx  dx  dx 

The  derivative  of  the  product  of  any  finite  number  of  factors 
is  equal  to  the  sum  of  the  products  obtained  by  multiplying 
the  derivative  of  each  factor  by  all  the  other  factors. 

Ex.  Show  that  the  operators  A  and  —  are  not  distributive  as  to 
multiplication.  '  ■*" 

27.  Differentiation  of  a  quotient. 

Let  y=-^-^^ 

then  y  +  ^y=:.l(^±A^. 


54 


DIFFERENTIAL   CALCULUS 


[Ch.  II. 


f(x  +  Ax) 
Ay      <i)(x  +  Ax) 

Ax                   Ax 

_  0 (x)f(x  +  Ax)  —f(x) (f) (x  +  Ax) 
Ax  </>  (x)  (f){x  +  Ax) 

By  subtracting  and  adding  ^Qc)f(x)  in  the  numerator, 
this  expression  may  be  written 


<l>{x) 


\fCx  +  Ax)-f(x) 
Ax 


]  -fix) 


Ax  <f)  {x)  (f){x  +  Ax) 

Hence,  by  equating  limits, 

du^<f>(x)f'(x)-f(x)<f>'(x)_ 

dx  [<^(2;)]^ 

Another  form  of  this  result  is 


\^(x-\-Ax)  —  ^(x) 

Ax 


dx\v) 


dx        dx 


[Art.  9,  Ths.  8,  9. 


(6) 


The  derivative  of  the  quotient  of  two  functions  is  equal  to 
the  denominator  multiplied  hy  the  derivative  of  the  numerator 
minus  the  numerator  multiplied  hy  the  derivative  of  the  de- 
nominator^ divided  hy  the  square  of  the  denominator. 

28,  Differentiation  of  a  commensurable  power  of  a  function. 

Let  y  =  w",  in  which  m  is  a  function  of  x ;  then  there  are 
three  cases  to  consider. 

1.  w  a  positive  integer. 

2.  w  a  negative  integer. 

3.  w  a  commensurable  fraction. 


1.    w  a  positive  integer. 

This  is  a  particular  case  of  (4),  the  factors  u,  v,  w, 

being  equal.     Thus 

dy         __i  du 
ax  dx 


all 


27-28.]    DIFFERENTIATION  OF  ELEMENTARY  FORMS  55 

2.  n  a  negative  integer. 

Let  n=  —  7n,  in  which  w  is  a  positive  integer ;  then 

"^^  l=^^'t    by  (6),  and  Ca^e  (1) 

-m-l  du 

=  —  mu        — ; 
dx 

1  dy  __i  du 

hence  -^  =  mi     — — 

dx  ax 

3.  w  a  commensurable  fraction. 

Let  n  =  - ,  where  jt?,  q  are  both  integers,  which  may  be 

either  positive  or  negative  ;  then 

p 
t/  =  u''  =  u^; 

hence  «/'  =  n^, 

and*  ^(^^)  =  f(^u'y, 

ax  dx 

I.e.,  qt/"  ^-f-=pu''  ^-— . 

ax  dx 

Solving  for  the  required  derivative, 

dy  _  p   P—idu  ^ 
dx      q         dx' 

hence  ^!*!!  =  nw"-*— .  (6) 

dx  dx 

The  derivative  of  aiiy  commensurable  power  of  a  function 
is  equal  to  the  exponent  of  the  power  multiplied  by  the  power 
with  its  exponent  diminished  by  unity,  multiplied  by  the  de- 
rivative of  the  function. 

♦  K  two  functions  be  identical,  their  derivatives  are  identical. 


56  DIFFERENTIAL   CALCULUS  [Ch.  II. 

These  theorems  will  be  found  sufficient  for  the  differenti- 
ation of  any  algebraic  function  ;  as  such  functions  are  made 
up  of  the  operations  of  addition,  subtraction,  multiplication, 
division,  and  involution,  in  which  the  exponent  is  an  integer 
or  commensurable  fraction. 

The  following  examples  will  serve  to  illustrate  the  theo- 
rems, and  will  show  the  combined  application  of  the  general 
forms  (1)  to  (6). 

ILLUSTRATIVE  EXAMPLES 

.  3  x2  -  2     „    -  </y 

1.    y  = — :  find  -^• 

■^         X  +  I    '  (Ix 


(x  +  l)4'  (3  z2  -  2)  -  (3  x2  -  2)  ^  (x  +  1) 
ay ax ax ^_ 

dx~  (x  +  1)'^ 


by  (6) 


£(3  .-2)  =£(3.-^)  -£(2)  by  (2) 

=  6x.  .  by  (1),  (6),  Ex.  7,  p.  49. 

£(.+  l)=|  =  l.  by(2) 

Substituting  these  results  in  the  expression  for  y^» 

dy  _  (x  +  1)  6  X  -  (3  x''  -  2)  _  3  x^  +  0  x  +  2 
dx  ~  (x  +  ly  ~  ~~(x  +  1)2 

2.   M  =  (3  s2  +  2)  VI  +  5s2;  find  — • 

ds 

du 


^  =  (3  s2  +  2)  -  VI  +  5s2  +  Vl  +  5  s2 .  !L  (3  s^  +  2).  by  (3) 

ds  ds  ds 

—  Vl  +  5  s2  =  S^^  (1+5  s2)2  / 

ds  ds  ^  ^ 

=  |(l  +  5.s-2)-i|(i+5,2)         by((J) 
5  s 


Vl  +  5  s2 


|-(3s2  +  2)=6s.  by  (6) 

as 


28.]         DIFFERENTIATION  OF  ELEMENTARY  FORMS  57 

Substituting  these  values  in  the  expression  for  — -> 

as 


.  VI  +a;2  +  Vl  -  x2  dy 

3.   y  =     ,  ; ;  find  -^ 

Vl  +  x2  -  Vl  -  a:2  f'a: 

First,  as  a  quotient,  by  (5),  J-^  =  (vT+x^-  vT^^)  ~(Vl  +  ^+  VT^^O 


(VT  +  Z2-Vl-Z2)2 


-  (  VH- a;2  +  >/r:^)  —  (  vT+72  -  VI  -  x2) 


(Vl   +x2-Vl  -X2)2 

— ( vrT^2  +  vr^^2)  ^  A  vrr^  +  —  vn^^.    by  (2) 

dx  dx  dx 

4-  VrT^^=  ^  (1  +  a;2)5  =  i  (1  +  a:2)-i  A(l  +  ^2).        by  (6) 
dx  dx  2  ax 

-^  (1  +  x2)  =  2  X.  by  (2)  and  (6) 

dx 

Similarly  for  the  other  terms.     Combining  the  results, 

dx         x3    V  Vl  -X*' 

Ex.  3  may  also  be  worked  by  first  rationalizing  denominator. 


EXERCISES 
Find  the  x-derivatives  of  the  functions  in  1-10. 

T^x 
2.    y  =      .    '''        •  6.   y  = 


Va2  -  x2  (  I  -t-  Vl  -  x* 

3.  !j  = —  •  -— — r--  7.    M  =  (2  rti  +  xi)  Vo^^^xi. 

(n  +  x)"     (o  +  x)"  ^       V  /   » 

4.  y  =    ^^  +  ^-  8.   y  =  (X  -  fl)(x  -  ft)(x  -  c)2. 

Va  +  Vx 


58  DIFFERENTIAL   CALCULUS  [Ch.  II. 


9.  y=jnzz:  10.  y=-"+i 


x"  -  1 

11.  Given,  (a  +  x)*  =  a^  +  5  a*x  +  10  a^x^  +  10  a^x^  +  5  ax*  +  x*;  find 
(a  +  x)*  by  diiferentiation. 

12.  Show  that  the  slope  of  the  tangent  to  the  curve  y  =  x^  is  never 
negative.     Show  where  the  slope  increases  or  decreases. 

13.  Given  hH^  +  aV  =  ^252^  find  ^  :  (1)  by  differentiating  as  to  x; 

dx 
(2)  by  differentiating  as  to  ^;  (3)  by  solving  for  y  and  differentiating 
as  to  X. 

14.  Show  that  form  (1),  p.  51,  is  a  special  case  of  (3). 

29.  Elementary  transcendental  functions.  Functions  that 
involve  operations  other  than  addition,  subtraction,  multi- 
plication, involution  (with  integer  exponent),  and  evolution 
(with  integer  index)  are  transcendental  functions  [Art.  4]. 

The  most  elementary  transcendental  functions  are  : 

Simple  exponential  functions,  consisting  of  a  constant 
number  raised  to  a  power  whose  exponent  is  variable, 
as  4%  a^; 

general  exponential  functions,  involving  a  variable  raised 
to  a  power  whose  exponent  is  variable,  as  2;*'"^; 
the  logarithmic  *  functions,  as  logo  ^i  logft  w ; 
the  incommensurable  powers  of  a  variable,  as  x-^,  u^ ; 
the  trigonometric  functions,  as  sin  w,  cos  u ; 
the  inverse  trigonometric  functions,  as  sin"'^,  tan~^2;. 

There  are  still  other  transcendental  functions,  but  they 
will  not  be  considered  in  this  book. 

The  next  four  articles  treat  of  the  logarithmic,  the  two 
exponential  functions,  and  the  incommensurable  power. 

*  The  more  general  logarithmic  function  log„  u  is  not  classified  separately. 

as  it  can  be  reduced  to  the  quotient  i^^iif. 

log,© 


28-30.]       DIFFERENTIATION  OF  ELEMENTARY  FORMS        59 


30.  Differentiation  of  log^x  and  log„u. 

Let  y  =  log„  X, 

then  y-V^y  =  log«  (a;  +  Aa;) 

Ay  ^  logg  (a:  +  Aa;)  -  log^  a; 
Ax  Aa;  ' 


1  1       /'a;  +  Aa;\ 


For  convenience  writing  A  for  Aa;,  and  re-arranging, 
Aa;      X   h         \       x. 


X         \        xj 


^  _1   lim 

dx~x^^^ 


logall+f" 


To  evaluate  the  expression  ( 1  +  - )    when  A  =  0,  expand  it 

by  the  binomial  theorem,  supposing  -  to  be  a  large  positive 
.   ,  h 

integer  m. 

The  expansion  may  be  written 

mj  m         1-2       m"  1-2.3 

which  can  be  put  in  the  form 


A       1\"»  _ .      .      1 V        mJ      1  \        7n/  V        mJ 
V^m)    --l  +  ^  +  i        2       +1        2  3 


+ 


1    2 


Now  as  w  becomes  very  large,  the  terms  —,  — ,  •  •  •  become 
very  small,  and  when  ?n  =  qo  the  series  becomes 


60  'DIFFERENTIAL   CALCULUS  [Ch.  U. 

The  numerical  value  of  the  sum  of  this  series  can  be 
readily  calculated  to  any  desired  approximation.  This  sum 
is  an  important  constant,  which  is  denoted  by  the  letter  «, 
und  is  equal  to  2.7182814  ...,  thus 

Jim  A  +  1  V=  g  =  2.7182814  ....* 

m  =  CO  \^         n/ 

The  number  e  is  known  as  the  natural  or  Naperian  base ; 
and  logarithms  to  this  base  are  called  natural  or  Naperian 
logarithms.  Natural  logarithms  will  be  written  without  a 
subscript,  as  log  x ;  in  other  bases  a  subscript,  as  in  log^  a;, 
will  generally  be  used  to  designate  the  base  ;  but  the  common 
logarithm,  logj^  x^  is  often  written  Log  x.  The  logarithm  of 
e  to  any  base  a  is  called  the  modulus  of  the  system  whose 
base  is  a. 

If  the  value,  ;^!^o(l  +  -  ]^  =  e,  be  substituted  in  the  ex- 
pression for  -f-^  there  results  [Th.  6,  p.  81 ;  Ex.  p.  29. 
ax 


dx       X 
More  generally,  by  Art.  21, 


logafi. 


^l.g.«  =  l.|.g,e.^.  (7) 

In  the  particular  case  in  which  a  =  e, 

The  derivative  of  the  logarithm  of  a  function  is  the  product 
of  the  derivative  of  the  function  and  the  modidus  of  the  si/stem 
of  logarithms,  divided  by  the  function. 

*  This  method  of  obtaining  e  is  rather  too  brief  to  be  rigorous  ;  it  assumes 

that  —  is  a  positive  integer,  but  that  is  equivalent  to  restricting  Ax  to 

Ax 
approach  zero  in  a  particular  way.     It  also  applies  the  theorems  of  limits  to 
the  sum  and  product  of  an  infinite  number  of  terms.    The  proof  is  completed 
on  p.  315. 


30-32.]      DIFFERENTIATION   OF  ELEMENTARY  FORMS         61 

31.  Differentiation  of  the  simple  exponential  function. 
Let  2/  =  a"  ; 

then  log  y  =  u  log  a. 

Differentiating  both  members  of  this  identity  as  to  a;, 

- 1^  =  log  «  •  T^'    by  form  (8), 
1/ax  dx 

dy      1  du 

therefore  ^""  ^ '"^"■"'' '  ^'  (*) 

The  derivative  of  an  exponential  function  with  a  constant 
base  is  equal  to  the  product  of  the  function^  the  natural  loga- 
rithm of  the  base,  and  the  derivative  of  the  exponent. 

32.  Differentiation  of  the  general  exponential  function. 

Let  y  =.  u", 

in  which  w,  v  are  both  functions  of  x. 

Take  the  logarithm  of  both  sides,  and  differentiate ;  then 

log  2/  =  vlogM, 

l|^  =  *logu  +  ^fi,byforms(S),(8), 
ydx       dx  udx 

dy  _     r,  ,  ^   I   ^  ^**"1 . 

dx  L  dx      II  dxj ' 

therefore  ^•'•  =  "l'"*"' i  +  Si]"  <»' 

The  derivative  of  jan  exponential  function  in  which  the  base 
is  also  a  variable  is  obtained  by  first  differentiating,  regarding 


62  DIFFERENTIAL   CALCULUS  [Ch.  II. 

the  base  as   constant,  and,  again,  regarding  the  exponent  as 
constant,  and  adding  the  results. 

33.   Differentiation  of  an  incommensurable  power. 

Let  y  =  M», 

in  which  n  is  an  incommensurable  constant ;  then 

log?/  =  wlogw, 

1  d^  _  n     du 
y  dx       u     dx 

dy  y     du 

dx  u    dx 

d    „  „_i  du 

-—u"=  nvr  '— — • 
dx  dx 

This  result  is  of  the  same  forln  as  (6),  so  that,  in  the 
theorem  of  Art.  28,  the  qualifying  word  "  commensurable  " 
can  now  be  omitted. 

Ex.      V  =  (4  x2  -  7)2+^-^,  find  ^ 

dx 


log  ?/  =  (2  +  V^^TTs)  log  (4  a:2  -  7). 

l^  =  ^^__log(4a;2-7)  +  (2+\/^2^^)_^^ —      [Art.  32. 
y  dx      yjx'i-  —  5  4x^  —  7 

^  =  r4x2  -  7^2+^'^  X  nog(4x2-7)      8(2+^x2-5)1 
dx      ^  '  L     Vx2"^^  4x2-7       J 

The  following  exercises  relate  to  the  differentiation  of 
combinations  of  algebraic,  logarithmic,  and  exponential 
functions. 

EXERCISES 
Find  the  x-derivatives  of  the  following  functions : 

1.  y  =  log (4x2  —  7x4-2).  4.  y=x"logx. 

2.  y=e«*+*.  5.   j'=  Vx  -  log(Vx  +  1). 

3.  w=c'+*.  6.   v= — "^ • 


32-36.]     DIFFERENTIATION   OF  ELEMENTARY  FORMS  63 

7.  y=-^^.  "•   y=«''- 

^  +  *'  12.   y=  log,  (3  a;2  _  V2T^). 

8.  y=e-(l-ar8).  ^^    ^^_^ 

9.  y=  log  (log  a:).  ^^  (or-l)^ 
10.   y  =  e''.                                                            (x  -  2)^(a;  -  3)^ 

In  14,  take  the  logarithm  of  both  members  before  differentiating. 

Articles  34-40  will  treat  of  the  differentiation  of  the 
Trigonometric  Functions  within  the  range  of  continuity. 

34.  Differentiation  of  sin  u. 
Let  y  =  sin  u, 

f h  ti  ^  —  '^^"  (^  +  ^fP  —  ^^rt  u     Am 

Ax  Am  Ax 

_  2  cos  ^  (2  M  +  Am)  sin  |^  Au     Am 
Am  Aa; 

^     ,   1  A    \     sin  i  A?t     Am 

=  cos  (m  4-  A-  Am  )  •  2 .  — . 

^        ^       ^        ^Am        Aa: 

but,  when  Am  =  0,  cos  (m  +  |  Am)  =  cos  m,  by  Art.  14,   and 
2 =^  1,  by  Art.  13 ;  hence,  passing  to  the  limit 

-^  gin  u  =  cos  t*  $^.  (12) 

due  dx 

The  derivative  of  the  sine  of  a  function  is  equal  to  the  prod- 
uct of  the  cosine  of  the  function  and  the  derivative  of  the 
function. 

35.  Differentiation  of  cos  u. 


Let  y  =  cosM  =  sin(^  — m], 


then     ^  =  ±^in('^-u\=coJ^-u\±('^-u\^. 
dx      dx       V2        )  V2        )du\^        Jdx 


^co8U  =  -sint«$^.  (18) 

doc  dx 


64  DIFFERENTIAL   CALCULUS  [Ch.  II. 

The  derivative  of  the  cosine  of  a  function  is  equal  to  minus 
the  product  of  the  sine  of  the  function  and  the  derivative  of  the 
function. 

36.  Differentiation  of  tan  u, 

sinw 


Let  y  =  tan  u  = 


COSM 


d    ■  .  d 

cos  W  •  sill  W  — SHIM-  COSM 

then  ^  = 't _ '^ by  (5) 

dx  cos"^  u 


9        du  ,     •  o        du        du 


COS^  M  •    ~r-  +  Sill'*  U 

dx  dx         dx 


-o— .     (12),  (13) 
COS''  u  cos^  u 

that  is,  -^taiiw  =  sec2u'**^.  (14) 

The  derivative  of  the  tangent  of  a  function  is  equal  to  the 
product  of  the  square  of  the  secant  of  the  function  and  the 
derivative  of  the  function. 

37.  Differentiation  of  cot  u, 

1 


Let  y  =  cot  u  = 


taiiw 


,,  dy        —1      d  .  sec^udu       ,_^   .^.^ 

then  -^  =  _-_._tan?t  =  --— ^— ,      (5),  (14) 

dx      tan^w   dx  tan^w  dx 

-^  cott*  =  -  csc2  u  ^.  (15) 

dx  dx 

The  derivative  of  the  cotangent  of  a  function  is  equal  to 
minus  the  product  of  the  square  of  the  cosecant  of  the  func- 
tion and  the  derivative  of  the  function. 


35-40.]     DIFFERENTIATION   OF  ELEMENTARY  FORMS  65 

38.   Differentiation  of  sec  u. 
Let  y  =  sec  M 


COSM 


, ,  dy        —\        d  sin  u  du 

then  -f-  =  — —  •  —  cos  M  =  ■ — —  — . 

ax      cos^  u     ax  cos^  u  ax 

-^secu  =  tan  usee u~-  (16) 

dx  doc 

The  derivative  of  the  secant  of  a  function  is  eqval  to  the 
product  of  the  secant  of  the  function,  the  tangent  of  the  func- 
tion, and  the  derivative  of  the  function. 

39.   Differentiation  of  esc  u. 

1 


Let  y  =  CSC  u=  — 


sin  w 

, ,  dy       —  1        d    ■  cos  u  du 

then  --^  =  -^-—  .  _sinw  =  -^--— , 

dx      sin'^  u     ax  sm^  w  dx 

^CSCt«=-CSCUCOtM  ^.  (17) 

die  dx 

The  derivative  of  the  cosecant  of  a  function  is  equal  to 
minus  the  product  of  the  cosecant  of  the  function,  the  cotan- 
gent of  the  function,  and  the  derivative  of  the  function. 

40.  Differentiation  of  vers  u. 

Let  g  =  vers  u  =  l  —  cos  m, 

,,  dy  d 

then  ,   =  — T~  ^^'^  ^' 

dx  dx 

#Yer8M  =  8iiiw^.  (18) 

da>  dx 

The  derivative  of  the  versed-sine  of  a  function  is  equal  to 
the  product  of  the  sine  of  the  function  and  the  derivative  of 
the  function. 


66  DIFFERENTIAL   CALCULUS  [Ch.  II. 

The  following  exercises  relate  to  the  differentiation  of 
combinations  of  algebraic,  logarithmic,  exponential,  and 
trigonometric  functions. 

EXERCISES 

Find  the  ^-derivatives  of  the  following  functions : 

1.  sin  5  x^.  5.   tan  a^  9.  tan  x  —  x. 

2.  sin^Ta:.  6,    log  tan  (^  x  +  |  tt)  .  10.  ain  (u  +  b)  cos  (u  —  b). 

3.  I  tan^ X  —  tan  a:.  7.   log  cot  x.  11.  x«'"*. 

4.  2sinxcosx.  8.   sinrixsin"x.  12.  sin  (sin  u). 


DIFFERENTIATION  OF   THE   INVERSE   TRIGONOMETRIC 
FUNCTIONS 

41.  Differentiation  of  sin-i  u. 

Let  1/  =  sin~i  u  ; 

then  sin  y  =  u; 

and,  by  differentiating  both  members  of  this  identity, 


hence 


I.e. 


cos 

dy 
^  dx 

_  du  ^ 
dx 

dx 

1 
cos^ 

du  _ 
'  dx 

1^ 

du 

±Vi 

— 

^m^ydx 

d_ 

sin 

-iw  = 

■*vi 

1 

du 
'dx 

dx 

1    -  «2 

- 

The  ambiguity  of  sign  accords  with  the  fact  that  sin~iM 
is  a  many-valued  function  of  m,  since,  for  any  value  of  u  be- 
tween —  1  and  1,  there  is  a  series  of  angles  whose  sine  is  u ; 
and,  when  u  receives  an  increase,  some  of  these  angles  in- 
crease and  some  decrease  ;  hence,  for  some  of  them, ; 

du 

is  positive,  and  for  some  negative.      It  will  be  seen  that, 
when  sin~^  u  lies  in  the  first  or  fourth  quarter,  it  increases 


40-42.]      DIFFERENTIATION  OF  ELEMENTABY  FORMS         67 

with  u,  and,  when  in  the  second  or  third,  it  decreases  as  u  in- 
creases. Hence,  if  it  be  agreed  that  sin~i  u  shall  mean  the 
angle  between  —  |  tt  and  +  ^  tt,  whose  sine  is  m,  then 

-^sin-iw  =  + 1 ,    :f8in->u  =  +       ^        g^.     (19) 

Thus  the  ambiguity  in  the  derivative  is  removed  by  speci- 
fying tliat  sin~i  u  is  to  mean  the  numerically  smallest  angle 
whose  sine  is  u. 

It  is  well  to  note  the  distinction  between  an  ambiguous 
derivative  and  a  non-unique  derivative.  In  the  present  case, 
the  ambiguity  disappears  when  any  particular  branch  of  the 
many-valued  function  is  specified,  and  thus  each  branch  has 
a  unique  derivative. 

The  derivative  of  the  anti-sine  of  a  function  i»  equal  to  the 
derivative  of  the  function  divided  by  the  square  root  of  unity 
minus  the  square  of  the  function. 

42.   Differentiation  of  cos-'m. 

It  may  be  proved,  by  the  method  used  in  Art.  41,  that 

d  ,  _         1         C?M 

—  cos  ^tt  =  T  — -• 

dx  Vl  —  u^  dx 

To  discriminate  between  the  two  values  of  this  derivative, 
observe  that,  when  cos~^  u  lies  in  the  first  or  second  quarter, 
it  decreases  as  u  increases,  and  when  in  the  third  or  fourth, 
it  increases  with  u.  Hence,  if  it  be  agreed  that  cos"^  u  shall 
mean  the  angle  between  0  and  tt,  whose  cosine  is  m,  then 

4co8-in=       -^      ,   Acos-»w  =  — ^l=r^.  (20) 

du  Vl  —  u^     ^^  Vl-u2  dx 

Here  the  ambiguity  in  the  derivative  is  removed  by  speci- 
fying that  cos~iw  is  to  mean  the  smallest  positive  angle 
whose  cosine  is  u. 

DIFF.  CALC. 6 


68  DIFFERENTIAL   CALCULUS  [Ch.  II. 

The  derivative  of  the  anti-cosine  of  a  funcfioh  is  equal  to 
minus  the  derivative  of  the  function  divided  by  the  square  root 
of  unity  minus  the  square  of  the  function. 

43.  Differentiation  of  tan-^w. 
Let  y  =  tan~^  u ; 

then  tan  y  =  u^ 

9    dy       du 
sec^y-f-  =  ——> 
dx       ax 

dy  _      \     du  _  1  du 

dx      sec^  y  dx       1  +  tan^  y  dx 

therefore  #  tan- » u  =  -^—  ^ .  (21 ) 

The  absence  of  ambiguity  accords  with  the  fact  that,  on 
each  of  its  branches  corresponding  to  the  same  value  of  m, 
tan^^w  is  an  increasing  function  of  u.  Unless  otherwise 
stated,  tan~^  u  is  specified  to  mean  the  numerically  smallest 
angle  whose  tangent  is  u. 

The  derivative  of  the  anti-tangent  of  a  function  is  equal  to  the 
derivative  of  the  function  divided  by  unity  plus  the  square  of 
the  function. 

44.  Differentiation  of  cotr'w. 

It  may  be  proved,  by  the  method  used  in  Art.  43,  that 

^cot-i«*  =  -^^.  (22) 

dx  1  +  u^dx 

On  each  of  the  branches  corresponding  to  the  same  value 
of  M,  cot~^  M  is  a  decreasing  function  of  u.  Unless  otherwise 
stated,  cot~^  u  is  specified  to  mean  the  numerically  smallest 
angle  whose  cotangent  is  u. 


42-46.]       DIFFERENTIATION   OF  ELEMENTARY  FORMS        69 

The  derivative  of  the  anti-cotangent  of  a  function  is  equal  to 
minus  the  derivative  of  the  function  divided  by  unity  plus  the 
square  of  the  function. 

45.  Differentiation  of  sec~*  u. 

Let  y  —  sec~*  t*, 

then  sec  y  =  u^ 

.         dy     du 

sec  y  tan  y-f-  =  -rr-~> 

dx     dx 

dv  \  du  1  du 


dx     sec y  tun  ydx     secy^/sec^y —  1  ^^ 

A.  sec-i  u  = ^ ^.  (28) 

**^  u^u^  _  1  <*a5 

K  it  be  agreed  that  8ec~*  u  shall  stand  for  the  numerically  smallest 
angle  whose  secant  is  «,  —  that  is  to  say,  if  when  «  is  positive  sec-* « 
shall  be  taken  between  0  and  ^tt,  and  when  «  is  negative  sec-^ti  shall  be 
taken  between  —  \ir  and  —  tt,  —  then  it  will  be  seen  on  comparing  the 
directions  of  algebraic  increase  of  u  and  sec-*  u  that  the  positive  sign 
should  be  given  to  the  radical  in  (23). 

The  derivative  of  the  anti-secant  of  a  function  is  equal  to  the 
derivative  of  the  function  divided  hy  the  product  of  the  function 
and  the  square  root  of  the  square  of  the  function  less  unity. 

46.  Differentiation  of  csc~*  u. 

It  may  be  proved,  by  the  method  of  Art.  45,  that 

A.  csc-i  u  = ^ ^.  (24) 

<*^  »>/«*«  -  1  ^^ 

Ex.  Show  that  the  algebraic  sign  is  correct  if  it  be  agreed  that 
csc-*«  shall  mean  the  numerically  smallest  angle  whose  cosecant  is  «. 

The  derivative  of  the  anti-cosecant  of  a  function  is  equal  to 
minus  the  derivative  of  the  function  divided  by  the  product  of 
the  function  and  the  square  root  of  the  square  of  the  function 
less  unity. 


70  DIFFERENTIAL   CALCULUS  [Cii.  II. 

47.   Differentiation  of  vers"'  u. 

Let  1/  =  vei\s~^  w; 

then  vers  i/  =  u, 

du      du 
sin  y-^  =  — -1 

dx      dx 

dy  _     1      dn  _  \  du 

dx      amy  dx      Vl  —  ( 1  —  vers  y)^  ^^' 

JL  vei-s-'  u  =         ^  ^.  (26) 

doc  V2  It  -  m'^  f^^ 

Ex.  Show  that  the  sign  of  the  radical  is  to  be  taken  positive  if 
vers"'  u  be  specified  to  mean  the  smallest  positive  angle  whose  versed- 
sine  is  u. 

The  derivative  of  the  anti-versed -sine  of  a  function  is  equal 
to  the  derivative  of  the  function  divided  by  the  square  root  of 
twice  the  function  minus  the  square  of  the  function. 

EXERCISES 
Differentiate  the  following  expressions : 

1.  arsin-'x.  7.   log(cos-' j:).  13.   sin  log ar. 

2.  tan  x  tan-' x.  8.   sin-^'iar*.  -^^    log  sin  a:. 

3.  sin-i^-:^!.  9.   vers-i^.  15.    V^HT^. 

4.  tan-.-^.  ■     10-   -^-'(-•^-5)-  ^^r\ 

1+^'  11.   sec-'        1       ■ 

5.  tan-'C.  Vl  -a;2  17.    cf"    «. 

6.  cos ~ '(log x).  12.   csc-'SVx.  18.  sin  (cos  x). 

The  results  of  this  chapter  are  for  convenience  summarized 
on  pages  71,  72 ;  they  will  suffice  to  differentiate  any  combi- 
nation of  algebraic,  logarithmic,  exponential,  trigonometric, 
and  inverse  trigonometric  functions. 


47. J          DIFFERENTIATION  OF  ELEMENTARY  FORMS          71 

d(c»)  =  c^.                                          (1) 

die  doc 

^iu  +  v  +  w)=f^,^  +  f^.  (2) 

dx  dx     dx     dx 

d(uv)  ^u^+v^.                            («) 

dx  dx        dx 

4-{uvtv)  ^uv^  +  uiv^  +  vw*^.      (4) 

dx  dx           dx           dx 

d  u  _     dx        dx  ^                           (6) 

dxv  v^ 

-f-u^  =nu^^—'                                (6) 

dx  fix 

^lOgeU  =1*IMl,                                                      (8) 

dx  udx 

A_«  _.              «    du 


dx"" 


=  log^«a«-^*  (.9) 


? 


ax  dx 

-^slnu  =eosu^.                             (12) 

da;  da? 

^cosM  =_sinM^.                            (18) 

da;  die 

^  tanu  =8ec2M^.                               (14) 

da?  dx 

Acotu  =-C8c2m^.                            (16) 

da?  da? 

^  sec  u  =  sec  w  tan  u  ^»                     (16) 

da?  da? 

-^  CSC  M  =  -  CSC  u  cot  M  ^ .                     (17) 

da?  dx     - 


72                               DIFFERENTIAL   CALCULUS  [Ch.  II. 

^TersM  =sinM^.                               (18) 

doc  doc                                  ^ 

d    .     ,  1        du                             ,^tks 

■^-  sin   '  M  =  — =^==r  ^-  •                               (19) 

aa?  Vl  -  m2  <?» 

^  c«s-' ««  =      ~^      ^**.                           (20) 

ace  Vl  -  m2  <*a5 

doc  1  +  ««2  ^^a;                               '^'^^ 

d            ,  —1       «fw                                           ,oav 

T-  cot  *  M  = :;  ^—  •  (22) 

doc  1  +  m2  doc 

<?         _1  —1           du                                  ,a^\ 

:;^CSC   1  «*  = ^^=r  ^— •  (24) 

doc  ^Vu-^l  *^^ 

<i            _i  1              du                                ,ac\ 

MISCELLANEOUS   EXERCISES 

In  Ex.  1-10  find  ^ : 
dx 

1.   y  =  log  (e^  +  e-').  5    «  =      ^ 

e'  -  l' 

2._y  =  I  -  I    •  6.  y  =  e-*'cosa;. 

3.    2/  =r  log  cot  X.  I.   y  -  JC          . 


8.   y  =  sec~^- 


4.  2/=(x-3)e2*  +  4xe»  +  x  +  3.  "^  2x^-1 

9.  y  =  sin  (2  «  —  7)  ;  «  =  log  x^.       10.   3/  =  e" ;  w  =  log'ar. 

11.   y  =  logs'  +  «•;«  =  sec<;  find  ^ 


^2-  ^  =  ";; ;;•     ^or  what  values  of  x  is  y  an  increasing  function  ? 

a^  +  X*  

13.  Prove  that  tan-^  ( +  ^   —    \  always  increases  with  x. 

14.  Show  that  the  derivative  of  tan-'  'v — ~  ^^^^  is  not  a  function  of  x. 

1  +  cos  X 

15.  Find  at  what  points  of  the  ellipse  —  +  ^  =  1,  the  tangent  cuts  off 
equal  intercepts  on  the  axes.  " 

16.  Find  -^  from  the  expression  xh/^  -  y^x^  +  6x^-5y  +  B  =  0. 

dx 


CHAPTER   III 

SUCCESSIVE  DIFFERENTIATION 

48.  Definition  of  nth  derivative.  When  a  given  function 
1/  =  <^{x)  is  differentiated  with  regard  to  x  by  the  rules  of 
Chapter  II,  then  the  result 

defines  a  new  function  which  may  itself  be  differentiated  by 
the  same  rules.     Thus, 


dx\dxj      dx 


The  left-hand  member  is  usually  abbreviated  to  — ^,  and 
the  right-hand  member  to  <^"(x);   thus, 

Differentiating  again  and  using  a  similar  notation, 

A(^)^^^cly"'(x-), 
dx  \dx^J      ds^ 

and  so  on  for  any  number  of  differentiations.  Thus  the 
symbol  — ^  expresses  that  y  is  to  be  differentiated  with 
regard  to  a;,  and  that  the  resulting  derivative  is  to  be  differ- 
entiated again ;    or,  in  other  words,  that  the  operation  — 

is  to  be  performed  upon  y  twice  in  succession.     Similarly, 

73 


74  DIFFERENTIAL   CALCULUS  [Oh.  111. 

— ^  indicates  the   performance  of   the   operation    —   three 
da^  dx 

times  upon  y,  and  so  on.     Thus  the  symbol  — ^  is  equiva- 

f  d\'  .  .       ^^" 

lent  to  ( —  ]  y.     It  is  called  the  nth  derivative  of  y  with 
\dxj 

regard  to  x. 

Ex.  If  t/  =  X*  +  sin  2x, 

^=43;8  +  2cos2x, 
dx 

^=12x2_4sin2a;, 

^  =  24x-8cos2z, 

^  =  24  +  16sin2z. 
dx* 


49.  Expression  for  the  nth  derivative  in  certain  cases.  For 
certain  functions,  a  general  expression  for  the  wth  derivative 
can  be  readily  obtained  in  terms  of  n. 

Ex.1.   If         y  =  .x;^  =  .x;  ^,=  e^;-;jf„  =  e", 

where  n  is  any  positive  integer.     If  y  =  e"*,  -—  =  a'c*". 

Ex.  2.   If  y  —  sin  x, 

-^  =  cos  a;  =  sin  (  x  +  -  ), 
dx  \        2/ 

^  =  cosfx  +  5)  =  sin(x  +  ^), 


dx2 


— ^  =  sni    X  H . 

If  «  =  sinax,  — ^=:  a"siu  I  ax  +  n— ). 

^  rfx»  V  2/ 


48-60.]  SUCCESSIVE  DIFFERENTIATION  75 

50.  Leibnitz's  *  theorem  concerning  the  »i,th  derivative  of  a 
product. 

Let  y  =  uVy  where  u,  v  are  functions  of  x  ;  then 

dy        dv   ,     du  ,  ,         du    dv 

-^  =  <A— -  +  V— ^  =  uv^  +  WjV,  where  — ,  — 
dx         dx        dx  dx   dx 

are  replaced  by  Mj,  v^  for  convenience ; 
again,      -—^  =  uv^  +  2  m^Vj  +  u^v. 

These  subscripts  and  coefficients  thus  far  foHow  the  same 
hiw  as  the  exponents  and  coefficients  of  tlie  binomial  series. 
To  test  wliether  this  law  is  true  universally,  assume  its  truth 
for  some  particular  value  of  w, 

d"y  ,  ,  n  I 

— ^  =  Mv„  +  nu^v„_i  +  •  •  • -— - .  UrV„_r 

dx"  (n  —  r)l  rl 

..  » 

w^+iv„_^i  H (1) 


(n  -  r  —  1)  !  (r  +  1  j  ! 


and  compare  the  result  of  differentiating  once  more  with  the 
result  of  changing  n  into  n  +  1  in  (1);  if  these  two  results 
are  the  same,  it  proves  that  if  the  law  be  true  for  any  one 
value  of  n,  it  will  also  be  true  for  the  next  liigher,  and  so 
on,  universally. 


•Gottfried  Wilhelm  Leibnitz  (1646-1716),  the  founder  of  the  nomencla- 
ture and  one  of  the  chief  foundere  of  the  philosophy  of  the  differential  calcu- 
lus. By  a  remarkable  coincidence  Sir  Isaac  Newton  (1642  O.S.-1727)  simul- 
taneously developed  the  same  science,  but  his  methods  and  notation  are 
somewhat  different.  For  the  history  of  this  remarkable  discovery,  see 
Cantor  :  Geschichte  der  Mathematik,  Vol.  8,  p.  1-50  ff. 


76  ~  DIFFERENTIAL   CALCULUS  [Ch.  III. 

By  differentiating  (1), 

(w  —  /•)  '  r . 

+  7 TtVt TTVT   l^r+l^'n-r  +  W^+2«'«-r-l|   + 

(w  —  r  —  1)  I  (r  +  1)  . 
=  ww«+i  +  (w  +  1)  MiV„  -1 


+  M^+iV„_, 


(n  -  r)  !  r !  "^  (w  -  r  -  1)  !  (r  +  1)  !  J  "^ 


+  7 ^\t}^\..,  '^r^rVn-r  +  -  ;  (2) 

(n  —  r)\  (r  +  1)1 

and  by  changing  n  into  w  +  1  in  (1),  the  result  is  seen  to  be 
the  same  as  (2). 

Now  the  expression  for  —^-  shows  that  the  law  is  true 

air 

for  w  =  2  ;  lience  it  is  universally  true,  and  thus  formula  (1) 
is  established. 

It  is  of   special  value  when   the   general   expression  for 
w„  and  for  w„  can  be  readily  obtained. 


Ex. 

Given 

y  =  x^e<"; 

find  ^. 
c?x'' 

Let 

u  =  x2, 

w  =  e«, 

then 

du  _ 
dx 

Uj  =  2  X, 

«2  =  2, 
«3  =  0, 

»3  =  a8e«», 

w«  =  0,   («  >  2),  «„  =  a»e<«. 
Substituting  these  values  in  (1), 

fin 

-—  (x^e")  =  x^a"e<"  4-  2  a»-ixe'"  +  n  C?i  -  1)  W-^e" 
dx»  ^  ^ 


50-51.]  SUCCESSIVE  DIFFERENTIATION  77 

51.  Successive  ic-derivatives  of  y  when  neither  variable  is 
independent.  Hitherto  the  differentiations  have  always  been 
performed  with  regard  to  the  independent  variable.  It  is, 
however,  sometimes  necessary  to  differentiate  a  function 
with  regard  to  a  variable  which  itself  depends  on  some  other 
variable.     Let  y  and  x  be  each  directly  given  as  functions 

of  an  independent  variable  t,  and  suppose  it  is  required  to 

dy    .  . 

express    -—    ni  terms  of  t. 

From  Arts.  21,  22, 

d^ 
dy  _dy    dt  _  dt  ^ 

dx      dt     dx      dx^  ^  ^ 

dt 

but  -^,  -TT  can  each  be  directly  expressed  in  terms  of  t  from 
dt     dt  ^      ^     . 

the  given  expressions  for  x,  y,  hence   j-  is  known  in  terms 
of  t. 

Thus  if  y  =  (^(0'  ^"<^  ^  =/(0'  ^^^^ 

d^V 
To  obtain  an  expression  similar  to  (1),  for  -v^,  it  may  be 

put  in  the  form 

d_^dy\ 

d^y  _  d  (dy\  _  d  fdy\    dt  _  dt  \dxj  ^o-v 

dx^      dx\dxj      dt\dxj    dx  rf^     ' 

di 

but,  by  differentiating  (1)  with  regard  to  ^, 

dx    d^y  _  dj£^    ^x 


d  (dy\_  dt    dt^      dt    dt^ 
t)~  fdx^ 

\dt) 


dtKdxJ  (dx^ 


78  DIFFERENTIAL   CALCULUS  [Ch.  III. 

hence,  by  (2),  J  = (3) 

\dt) 
Thus,  if  y  =  <^C0  ^"'^  ^  =f(^)i 

cPy_f'(t)<f>"(t)-<f>'(t)f'(t) 

Expressions  similar  to  (1)  and  (3),  but  more  complicated, 
can  be  obtained  for  the  higher  derivatives. 

Next  let  y  be  given  directly  in  terms  of  x^  and  x  in  terms 

of  t ;  then  -^  can  be  first  expressed  in  terms  of  a;,  and  the 

ax 

result  in  terms  of  t  by  elimination. 
Thus  if  y  =  4>(x\  x=f(t). 


then  ^=^'(x)=cf>'lfCt}l 


EXERCISES 

1.  «  =  x*  -  4  x8  +  6  x2  -  4  a:  +  1 ;     find  ^. 

2.  y  =  (x-  8)  e^  +  ixe'^  +  x;    find  — ^• 

3.  y  =  x6;      find  ^• 

r/x6 

4.  u  =  x^\osx;     find  — ^• 
^  ^  r/x< 

5.  V  =  log  (e'  +  6-==)  ;     find  ^• 
"  ^^  ^  dx« 


61.]  SUCCESSIVE  DIFFERENTIATION  79 

6.    v  =  —\o<fx  —  -:     faiid  — =^- 

*  ft        °  -..  //v'2 


7.  y  =  tan-  x  +  8  log  cos  x  +  3  z^  j     find      -^ 

8.  M  =  c"  sill  ftx ;    prove  '-^  —  2(i'-^  +  (n^  +  fr)  i/  -  0. 

ihr  ax 

9.  J  =  u  cos  (log  x)  +  6  sin  (log  x)  ;     prove  x^  ^  +  x  -^  +  ^  = 

10.  M  =  tail  X  +  sec  x ;     find  --^^ 

dx'^ 

11.  y  =  (x'  +  «•-')  tan-'  •*-  ;     find  j^- 

12.  1/  =  e  ^  cos  X ;     prove  — ^  +  4  y  =  0. 

13.  ,.j:z:;fi„d^. 

'X  —  «  rtX'' 

14.  ^  =  X''-'  log  X,  («  a  positive  integer)  ;    find  -p^- 

15.  y  =  log  ^-^^  ;     find  ^• 
■'  °  I  +  X  r/x» 

16.,  =  -^;     find^. 
1  —  X  dx* 

17.  M  =  sec  '2  X ;     find  — ^• 

18.  y  =  1  +  xe*;    find  ^  in  terms  of  y. 

dx'^ 

19.  M  =  tan  (x  +  v)  ;     find  —  in  terms  of  y. 

dx^ 

20.  f-\-y  =  x^;     find  ^^• 

21.  e'  +  x  =  e»  +  y;     find  ^• 

dx^ 

22.  e"  +  xif  -  ^   =  0  ;  find  '^• 

(fx^ 

23.  .y3  +  x3  -  3  nxy  =  0 :     find  '^ 


dx^ 


24.    M  = ^^ ;    find  '^■ 


80  DIFFERENTIAL   CALCULUS  [Ch.  III.  51. 

25.  y  =  ilog--±^;    find^. 

X  —  a  ax** 

26.  V  =  x-'  •  X ;     find  — ^• 
•^  rfx» 

27.  M  =  x2e«^ ;     find  ^• 

rfx" 

28.  y  =  x^  log  X ;     find  — -^• 

rfx" 

29.  y  =  T^;  fi"^?^- 

1  +  X  ax" 

30.  y^e'^sinx;  find  — -^• 

rfx" 

31.  Show  that  the  members  of  equation  (3),  p.  78,  become  identical 
when  I  is  replaced  by  x. 

32.  Replacing  t  by  y,  show  that 

(Py  _         dy^ 

\dy) 
Also  derive  this  relation  independently. 

33.  Verify  this  relation  when  y  =  sin  x. 

34.  Find  when  the  slope  of  the  curve  y  =  tan  x  increases  with  x;  and 
when  it  decreases  as  x  increases. 

35.  Show  that  the  slope  of  the  curve  y  =/(x)  changes  from  increas- 
ing to  decreasing  when  /"(x)  changes  its  sign.  Apply  to  the  cui'ves 
y  =  sinx,  y  =  sin^x. 


CHAPTER   IV 
EXPANSION  OF  FUNCTIONS 

52.  It  is  sometimes  necessary  to  expand  a  given  function 
in  a  series  of  powers  of  one  of  its  variables.  For  instance, 
in  order  to  compute  and  tabulate  the  successive  numerical 
values  of  sin  x  for  different  values  of  a:,  it  is  convenient  to 
have  sin  x  developed  in  a  series  of  powers  of  x  with  coeffi- 
cients independent  of  x. 

Simple  cases  of  such  development  have  been  seen  in 
algebra  ;  for  example,  by  the  binomial  theorem, 

(a  +  xy  =  a"  +  wa"-'a;  4- '^^'^~^^ a^-^x^  +—;  (1) 

and  again,  by  ordinary  division, 
1 


1- 


=  l+X+X^  +  3^+-'.  (2) 


It  is  to  be  observed,  however,  that  the  series  is  a  proper 
representative  of  the  function  only  for  values  of  x  within  a 
certain  interval  ;  for  instance,  it  is  shown  in  works  on  algebra 
that  when  n  is  not  a  positive  integer,  the  identity  in  (1) 
holds  only  for  values  of  x  between  —  a  and  4-  «,  and  that 
the  identity  in  (2)  holds  only  for  values  of  x  between  —  1 
and  +1.  In  each  case,  if  a  finite  value  outside  of  the 
stated  limits  be  given  to  x,  the  sum  of  an  infinite  number  of 
terms  of  the  series  will  be  infinite,  while  the  function  itself 
will  be  finite.     In  both  of  these  examples  the  stated  interval 

81 


82  DIFFERENTIAL   CALCULUS  ICu.  IV. 

of  equivalence  of  the  series  and  its  generating  function  is 
the  same  as  the  interval  of  convergence  of  the  series  itself. 
The  general  theory  of  the  convergence  and  divergence  of 
series,  so  far  as  necessary  for  the  present  purpose,  is  briefly 
outlined  in  the  next  two  articles. 

53.  Convergence  and  divergence  of  series.*  An  infinite 
series  is  said  to  be  convergent  or  divergent  according  as  the 
sum  of  the  first  n  terms  of  the  series  does  or  does  not 
approach  a  finite  limit  when  n  is  increased  without  limit. 

For  example,  the  sum  of  the  first  n  terms  of  the  geometric 

series 

a  +  ax-\-  aaP'  +  aa?  +  ••• 

\  —X 

First  let  x  be  numerically  less  than  unity ;  then  when  n 
is  taken  sufficiently  large,  the  term  a;"  =  0  ;  hence 

«„  = ,    when    n  =  cc. 

1  —  X 

Next  let  X  be  numerically  greater  than  unity  ;  then  when 
w  =  Qo  ,  a;"  =  00  ;  hence,  in  this  case 

8„  =  GO  ,    when    n  =  cc  . 

Thus  the  given  series  is  convergent  or  divergent  according 
as  X  is  numerically  less  or  greater  than  unity.  The  condi- 
tion of  convergence  may  then  be  written 

-1<3:<1, 

and  the  interval  of  convergence  is  between  —  1  and  +  1. 

*  For  an  elementary,  yet  comprehensive  and  rigorous,  treatment  of  this 
subject  see  Professor  Osgood's  "Introduction  to  Infinite  Series  "(Harvard 
University  Press,  1897). 


52-54.]  EXPANSION  OF  FUNCTIONS  83 

Similarly  the  geometric  series 

whose  common  ratio  is  —  82;,  is  convergent  or  divergent 
according  as  82;  is  numerically  less  or  greater  than  unity. 

The  condition  of  convergence  is  —  1  <  3  a:  <  1,  and  the 
interval  of  convergence  is  between  —  ^  and   +  ^. 

The  definition  just  given  and  illustrated  is  sometimes  more 
briefly  stated  as  follows  :  An  infinite  series  is  said  to  be  con- 
vergent or  divergent  according  as  the  sum  of  the  series  to 
infinity  is  finite  or  infinite. 

It  is,  however,  to  be  carefully  borne  in  mind  that  the 
phrase  ''  the  sum  of  the  series  to  infinity."  is  only  an  abbrevi- 
ation for  the  more  precise  phrase  "the  limit  approached  by 
the  sum  of  the  first  n  terms  when  n  is  made  larger  and  larger 
without  limit." 

54.  General  test  for  interval  of  convergence.  The  follow- 
ing summary  of  algebraic  principles  leads  up  to  a  test  that 
is  sufficient  to  find  the  interval  of  convergence  of  a  series  of 
the  most  usual  kind,  that  is,  a  series  consisting  of  positive 
integral  powers  of  x,  in  which  the  coefficient  of  x"  is  a 
known  function  of  n. 

1.  If  s„  is  a  variable  that  continually  increases  with  n, 
but  for  all  values  of  n  remains  less  than  some  fixed  number 
Jc,  then  8n  approaches  some  definite  limit  not  greater  than  k. 
[An  exercise  on  the  definition  of  a  limit.] 

2.  If  one  series  of  positive  terms  is  known  to  be  conver- 
gent, and  if  the  terms  of  another  series  be  positive  and  less 
than  the  corresponding  terms  of  the  first  series,  then  the 
latter  series  is  convergent.      [Use  1.] 

3.  If  after  a  given  term  the  terms  of  a  series  form  a 
decreasing  geometric  progression,  then  (a)   the  successive 

DIFF.   CALC.  7 


84  DIFFERENTIAL    CALCULUS  [Ch.  IV. 

terms  approach  nearer  and  nearer  to  zero  as  a  limit ;  and 
(6)  the  sum  of  all  the  terms  approaches  some  fixed  constant 
as  a  limit.     [Use  method  of  last  article.] 

4.  If  the  terms  of  a  series  be  positive,  and  if  after  a  given 
term  the  ratio  *of  each  term  to  the  preceding  be  less  than  a 
fixed  ijroper  fraction,  the  series  is  convergent.     [Use  2  and  3.] 

5.  If  there  be  a  series  A  consisting  of  an  infinite  number 
of  both  positive  and  negative  terms,  and  if  another  series  B, 
obtained  therefrom  by  making  all  the  terms  positive,  is 
known  to  be  convergent,  then  the  series  A  is  convergent. 

For  the  positive  terms  of  A  must  form  a  convergent  series, 
otherwise  the  series  B  could  not  be  convergent ;  similarly 
the  negative  terms*  of  A  must  form  a  convergent  series. 
Let  the  sums  of  these  convergent  series  be  w,  —  v.  Let  the 
first  n  terms  of  series  A  contain  m  positive  terms  and  p 
negative  terms ;  and  let  their  three  sums  be  respectively 
*S'„,  S,„,  —  Tp',,  then  /S'„  =  S,„  —  T^.  Now  when  n  =  cc,  so 
does  w  =  00,  and  p  ~  cc,  hence 

lim    o   _     lim    y    _    lim    nj      ■        .e  _  „  _  „ . 

therefore  the  series  A  is  convergent. 

Definitions.  The  absolute  value  of  a  real  number  x  is  its 
numerical  value  taken  positively,  and  is  written  \x\.  The 
equation  |  a;  |  =  |  a  |  indicates  that  the  absolute  value  of  x  is 
equal  to  the  absolute  value  of  a.  When,  however,  x  and  a 
are  replaced  by  longer  expressions,  it  is  convenient  to  write 
the  relation  in  the  form  a;  |  =  1  a,  in  which  the  symbol  |  =  |  is 
read  "equals  in  absolute  value."     Similarly  for  the  symbols 

l<M>l- 

Any  series  of  terms  is  said  to  be  ahsolutely  or  uncondition- 
ally convergent  when  the  series  formed  by  their  absolute 
values  is  convergent.     When  a  series  is  convergent,  but  the 


54.]  EXPANSION    OF  FUNCTIONS  85 

series  formed  by  making  each  term  positive  is  not  convergent, 
the  first  series  is  said  to  be  conditionally  convergent.* 

E.g.,i\\&  series  —  —  —  +  —  —  •••  is  absolutely  convergent;  but  the 
series  \  —  \  +  \  —  •"  is  conditionally  convergent. 

6.  If  there  be  any  series  of  terms  in  which  after  some  fixed 
term  the  ratio  of  each  term  to  the  preceding  is  numerically 
less  than  a  fixed  proper  fraction  ;  then, 

(a)  the  successive  terms  of  the  series  approach  nearer 
and  nearer  to  zero  as  a  limit ; 

(6)  the  sum  of  all  the  terms  approaches  some  fixed  con- 
stant as  a  limit ;  and  the  series  is  absolutely  convergent. 
[Use  3,  4,  5.] 

Ex.  1.   Find  the  interval  of  convergence  of  the  series 

1  +  2  .  2 X  +  3  •  4 z2  +  4  •  8 x3  +  .5  •  16 X*  +  .... 

Here  the  nth  term  is  ra2"-^x"-i,  and  the  (n  +  l)st  term  is  (n  +  l)2"x'», 

hence  »^  ^  (»  +  1)  2"x'>^  (»  +  l)2x^ 

M„  n  2"-i  x"-'  n 

therefore  when  n  =  oo,  — 2±i  i  2  x. 

It  follows  by  (6).  that  the  series  is  absolutely  convergent  when 
—  l<2x<l,  and  that  the  interval  of  convergence  is  between*—  \  and 
4-  \.  The  series  is  evidently  not  convergent  when  x  has  either  of  the 
extreme  values. 

Ex.  2.    Find  the  interval  of  convergence  of  the  series 

•      X  ofi  x^ £^  (—  1)"  ^x-"~^ 

1-3      3. 38  "^5-35     T^"*"  ■■*  ■*"  (2ii-l)3-'^"-i '^■'" 

*  The  appropriateness  of  this  terminology  is  due  to  the  fact  that  the  terms 
of  an  absolutely  convergent  series  can  be  rearranged  in  any  way,  without 
altering  the  limit  of  the  sum  of  the  series  ;  and  that  this  is  not  true  of  a  con- 
ditionally convergent  series.     Thus  the  sum  of  the  series ^  +  ^ •.•  is 

independent  of  the  order  or  grouping ;  but  the  sum  of  the  series 
1  —  J  +  J  —  ^  +  •..  can  be  made  equal  to  any  number  whatever  by  suitable 
re-arrangement.     [For  a  simple  proof  see  Osgood,  pp.  43,  44.] 


86  DIFFERENTIAL   CALCULUS  [Ch,  IV. 

i;^ '  - 1 2  n  +  1  ■  :5=^''+'  ■  x-^"-'  -  2  n  +  r  3^" 
lience  ^^^±1  ^  _    when  /»  i  xi ; 

thus  the  series  is  absolutely  convergent  when  — <  1,  i.e.,  when  —  3<a:<3, 

and  the  interval  of  convergence  is  fi-oni  —  3  to  +  3.    The  extreme  values 
of  X,  in  the  present  case,  render  the  series  conditionally  convergent. 


Ex.3.  Show  that  the  serie.  i(£)  -  1^ (i)' +!(£)' ->^(£)' 


+ 


has  the  same  intei'val  of  convei-gence  as  the  last ;  but  that  the  extreme 
values  of  x  render  the  series  absolutely  convergent. 

55.  Interval  of  equivalence.  Remainder  after  u  terms. 
Tire  last  article  treated  of  the  interval  of  convergence  of  a 
given  series  without  reference  to  the  question  whether  or  not 
it  was  the  development  of  any  known  function.  On  the  other 
hand,  the  series  that  present  themselves  in  this  chapter  are 
the  developments  of  given  functions,  and  the  first  question 
that  arises  is  concerning  the  interval  of  equivalence  of  the 
function  and  its  development. 

When  a  series  has  such  a  generating  function,  the  differ- 
ence between  the  value  of  the  function  and  the  sum  of  the 
first  n  terms  of  its  development  is  called  the  remainder  after  n 
terms.*  Thus  if  f{x)  be  the  function,  S„{x)  the  sum  of 
the  first  n  terms  of  the  series,  and  R„(^x)  the  remainder 
obtained  by  subtracting  SJx)  from  f(x),  then 

fix)  =  S^ix)  +  R„(x% 

in  which  >S'„(a;),  R„(x)  are  functions  of  n  as  well  as  of  x. 

*  In  some  discussions  of  convergence  of  series  without  any  reference  to 
a  generating  function,  the  phrase  "  remainder  after  n  terms"  is  occasionally 
used  in  a  sense  different  from  that  given  above,  which  is  the  recognized  usage 
in  treating  of  the  equivalence  of  a  function  and  its  development. 


54-50.]  .   EXP  AS  SIGN    OF  FUNCTIONS  87 

A  sufficient  conflitiou  for  the  convergence  of  the  series  is 
that  Rn(p^^  approach  a  finite  limit  when  n  =  ac  ;  for  in  that 
case     S„(x},  =f(x}—R„(a-),  =  ci  finite  number,  when  w  =  oc. 

Thus  the  interval  of  convergence  extends  over  those 
values  of  x  that  make  ^  V^^  Rnip)  equal  to  any  number  not 
infinite,  and /(a;)  itself  not  infinite. 

On  the  other  hand,  the  interval  of  equivalence  of  the 
series  and  its  generating  function  extends  only  over  those 
values  of  x  that  make  ^  ""^  R„(x)  =  0  ;  for  it  is  only  in  that 

case  that  S„(x)=f{x^,  when  n=-c. 

Thus  the  interval  of  equivalence  may  possibly  be  nar- 
rower than  the  interval  of  convergence. 

It  will  appear  later,  however,  that  in  the  case  of  all  the 
ordinary  functions,  ]^  Rni^)  ^^ill  be  zero  for  certain 
values  of  x^  and  infinite  for  all  other  values  of  x ;  and  that 
thus  the  intervals  of  convergence  and  of  equivalence  are 
identical. 

56.   Maclaurin*s  expansion  of  a  function  in  power-series.* 

It  will  now  be  shown  tliat  all  the  developments  of  functions 
in  power-series  which  were  studied  in  algebra  and  trigo- 
nometry are  but  special  cases  of  one  general  formula  of 
expansion. 

It  is  proposed  to  find  a  formula  for  the  expansion,  in 
ascending  positive  integral  powers  of  x,  of  any  assigned 
function  which,  with  its  successive  derivatives,  is  continuous 
in  the  vicinity  of  the  value  x  =  Q. 

♦Named  after  Colin  Maclaurin  (1698-1746),  who  published  it  in  his 
"Treatise  on  Fluxions"  (1742)  ;  but  he  distinctly  says  it  was  known  by 
Stirling  (1690-1772),  who  also  published  it  in  his  "  Methodus  DifEerentialis " 
(1730),  and  by  Taylor  (see  Art.  65). 


88  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

The  preliminary  investigation  will  proceed  on  the  hypoth- 
esis that  the  assigned  function  f(x)  has  such  a  development, 
and  that  the  latter  can  be  treated  as  identical  with  the 
former  for  all  values  of  x  within  a  certain  interval  of  equiva- 
lence that  includes  the  value  x  =  0.  From  this  hypothesis 
the  coefficients  of  the  different  powers  of  x  will  be  deter- 
mined. It  will  then  remain  to  test  the  validity  of  the  result 
by  finding  the  conditions  that  must  be  fulfilled,  in  order 
that  the  series  so  obtained  may  be  a  proper  representation 
of  the  generating  function. 

Let  the  assumed  identity  be 

f(x)  =  A+Bx+C2^  +  D2^  +  i:x*  +  •••,  (1) 

in  which  A,  B,  (7,  •••  are  undetermined  coefficients  indepen- 
dent of  X. 

Successive  differentiation  with  regard  to  x  supplies  the 
following  additional  identities,  on  the  hypothesis  that  the 
derivative  of  each  series  can  be  obtained  by  differentiating 
it  term  by  term,  and  that  it  has  some  interval  of  equivalence 
with  its  corresponding  function  : 

/(x)  =  B  +  2  Cx  -^S  J)x^  +  4  Ua^  -\-  —, 
f'Qx')  =2(7-f-3-2i>a;  +  4.3j;^+-, 
/"(a;)?=  3  .  22)  +  4  •  3  .  2  JKs  + -, 
/'v(a;)=4.3.2^+-.., 

in  which,  by  the  hypothesis,*  x  may  have  any  value  within 
a  certain  interval  including  the  value  x=  0. 

*  The  hypothesis  here  made  with  regard  to  each  sei'ies  would  not  be 
admissible  in  a  process  of  demonstration.  This  preliminary  investigation 
is  for  the  purpose  of  discovering  what  the  development  is,  if  any  exists.  The 
validity  of  the  development  is  fully  tested  in  Arts.  60-65. 

It  may  be  of  interest  to  refer  to  Professor  Osgood's  "  Introduction  to 
Infinite  Series,"  pp.  54,  fll,  for  a  proof  that  within  its  interval  nf  convergence 
a  power-series  is  a  continuous  and  differentiable  function  of  x,  and  that  its 


56.]  EXPANSION   OF  FUNCTIONS  89 

The  substitution  of  zero  for  x  in  each  identity  furnishes 
the  following  equations  : 

/(O)  =  A,  /(O)  =  B,  /"(O)  =2  0,  /"(O)  =  3  .  2  i>, ... ; 
hence  ^  =/(0),  ^=/(0),  C=-^^,  D=-(^^,-. 

The  unknown  coefficients  of  (1)  are  thus  expressed  in 
terms  of  known  indicated  operations ;  and  substitution  in 
(1)  gives  the  form  of  development  sought : 

Here  the  symbol  /"(O)  is  used  to  indicate  the  operation  of 
differentiating  /(a;)  with  regard  to  x,  n  times  in  succession, 
and  then  substituting  zero  for  x  in  the  expression  for  the  wth 
derivative. 

The  resulting  constant,  when  divided  by  w  !,  is  the  required 
coefficient  of  the  nth  power  of  the  variable  in  the  assumed 
development  of  the  function. 

It  remains  to  examine  what  are  the  conditions  that  must 
be  fulfilled  in  order  that  the  series  so  found  may  be  a  proper 
representative  of  the  function.  This  question  can  only  be 
fully  answered  when  the  expression  for  R„(x).,  the  remainder 
after  n  terms,  has  been  obtained.  This  expression  will  be 
derived  after  another  series,  which  may  be  regarded  as  a 
generalization  of  (2),  has  been  established. 

There  are,  however,  certain  preliminary  conditions  that 

true  derivative  can  be  obtained,  within  the  same  interval,  by  differentiating 
the  series  term  by  term. 

This  theorem  is,  however,  not  necessary  to  the  demonstration  of  Mac- 
lauriii's  or  Taylor's  theorem,  as  the  series  treated  in  Art.  60  consists  of  only 
a  finite  number  of  terms. 


90  BIFFEREI^TIAL    CALCULUS  [Ch.  IV. 

are  easily  seen  to  be  necessary  in  order  that  the  series  may 
give  the  true  value  of  the  function. 

First,  the  functions  f(x),  f'(x)^  f"(x),  •••  must  all  satisfy 
the  condition  of  being  continuous  in  the  vicinity  of  a;  =  0  ; 
otherwise  some  of  the  coefficients/(0;,/'(0),/"(0),  •••  would 
be  infinite  or  indeterminate,  and  the  series  would  have  no 
definite  sum  for  any  value  of  x^  showing  that  the  given  func- 
tion f(x)  could  have  no  development  in  the  form  prescribed. 

5  J 

Ex.  Show  that  the  functions  log  a;,  x^,  — cannot  be  developed  in 

powers  of  x.  e-'  +  1 

When  this  condition  is  satisfied,  it  is  further  necessary  for 
the  equivalence  of  the  function  and  its  development  that  the 
values  of  x  be  restricted  to  lie  within  a  certain  interval  not 
wider  than  the  interval  of  convergence  of  the  series. 

The  method  of  computing  the  coefficients  of  the  successive 
powers  of  x  in  tlie  development  of  a  given  function,  will  be 
illustrated  by  a  few  examples. 

Ex.  1.  Expand  sin  x  in  powers  of  x,  and  find  the  interval  of  conver- 
gence of  the  series. 

Here  J\x)  =  sin  x,  /(O)  =  0, 

f(x)  =  COHX,  /'(0)=1, 

/"(x)  =  -sina:,  /"(0)  =  0, 

/"(x)  =  -cosx,  /"'(0)=-l, 

/i^(x)  =  sinx,  /^^(0)  =  0, 

/v(x)  =  cosx,  r(0)  =  l, 


Hence,  by  (2), 

sin  X  =  0  +  1  •  X  +  0  •  a;2  _  l.x»  +  0-3:*  +  —s*  •-, 
3!  5! 

thus  the  required  development  is 

111  /_  TVn-l 

sinx  =  x  -  —x'  +  — a:6  -  — x'  + h  -i — -^ x^"  >  -f  .... 

3!  51  7!  (2n-l)! 


56.]  EXPANSION   OF  FUNCTIONS  91 

To  find  the  interval  of  convergence  of  the  series,  use  the  method  of 
Art.  54,  then 

«„  '  ~  '  (2  n  +  1)  !'  (2  «  -  1)  !  ~  (2  n  +  1)2  n 

and  this  ratio  approaches  the  limit  zero,  when  n  becomes  infinite,  how- 
ever large  be  the  constant  value  assigned  to  x.  This  limit  being  less 
than  unity,  the  series  is  convergent  for  any  finite  value  of  x,  and  hence 
the  interval  of  convergence  is  from  —  xi  to  +x. 

Assuming,  for  the  present,  that  the  value  x  =  .5,  for  example,  lies 
within  the  interval  of  equivalence  of  sin  x  and  its  development,  the 
numerical  value  of  the  sine  of  half  a  radian  may  be  computed  as  follows: 


sin  (.5)  =  .o  —  ^  -'^   '\ ^^ — '^ — ^^— 


2-3      2:3-4.5     2- 3-4. 5.6.7 
=  .5000000 

-  .0208333 
+  .0002604 

-  .0000015 
+  .0000000 

sin  (.5)=  .4794256... 

Show  that  the  ratio  of  Mj  to  u^  is  ^;  and  hence  that  the  error  in  stop- 
ping at  u^  is  numerically  less  than  u^  [jfy  +  (^y  +  •••]'<  sst  "<• 

Ex.  2.    Show  that  the  development  of  cos  x  is 

,      x^   ,  X*      x«   ,        ,  C— l)»-ix2»-2  , 

cos  X  =  1 -H h  ^ h  •••» 

2 !      4  !      6 !  (2  n  -  2) ! 

and  that  the  interval  of  convergence  is  from  —  oo  to  +  oo. 

Ex.  3.   Develop  the  exponential  functions  a^,  e^. 

Here 

f(x)  =  a',  f'{x)  =  a^  log  a,  f"{x)  =  rt-^(log  n)2  —f"(x)  =  n->'(]og  n)», 
hence       /(0)=  1,  /'(0)=  log«,  /"(*))=  Qoga^,  •.•/"(0)  =  (logn)«, 
and  «x  =  1  +  (log  a)x  +  HM^'z"  +  ...  +  (l2S^'^«  +  .... 

As  a  special  case,  putting  a  =  e,  the  Naperian  base, 
then  log  a  =  log  e  =  1, 

and  c^  =  1  +  x +  —  +  —  +••• +  —  +  .... 

2!      3!  n! 

These  series  are  convergent  for  every  finite  value  of  x. 


92  DIFFERENTIAL    CALCULUS  [Ch.  IV. 

Ex.  4.    Find  the  development  of  tan  x. 

Let        /(-i)  =  tan  x, 

then  /'(x)=  sec^a;, 

/" (a)  =    2  sec^ x  tan  x, 

f"  (r)  =    4  sec''^  x  tan^  x  +  2  sec*  a?, 

/•  v(a;)  =    8  sec^  x  tan^  x  +  16  sec*  x  tan  ar, 

p(x)-  16sec2xtan*a;  +  88  sec*  a;  tan^  x  +  16sec«x, 

/v>(x)  =  32  sec2  X  tan6  x  +  416  sec*  x  tau^  x  +  272  sec«  x  tan  x, 

/vn(a;)=64sec2xtan6x+1814sec*xtan*x  +  2880sec8xtan2x  +  272.sec8x,  •••. 

Hence       /(O)  =  0,  /'(O)  =  1,  /"(O)  =  0,  /'"(O)  =  2,  /'-(O)  =0, 

/v(0.)  =  16,  / V  (0)  =  0,  / V-  (U)  =  272, .. . 

o  16         '^7'^ 

therefore  tan  x  =  x  +  —  a;^  H — x^  +  ^^x"  +  ... 

3!         5!  71 

=  X  H — x"  H X*  H x^  +  .... 

3         15         315 

Here,  as  in  many  other  cases,  the  law  of  succession  of  the  coefficients 
is  very  complicated,  and  it  is  not  possible  to  express  the  coefficient  of  x" 
directly  in  terms  of  n.  Thus  the  interval  of  convergence  of  the  series 
cannot  be  obtained  by  simple  methods. 

Ex.  5.   Develop  c^"*  in  powers  of  x. 

Let         /(x)=e«n^, 
then  /'  (x)  =  gsin  ^  cos  x, 

f"(x)  =  es'n''(cos2x  -  sin  x), 

/'"(x)  =  e*'"-^(cos^  X  —  3  sin  x  cos  x  —  cos  x), 

/iv^j.)  —  gsin  r^-pos*  X  —  6  sin  xcos^x  —  4cos2x  +  Ssin^x  +  sin  x), 

/V(x)=e8in-^ 

(cos*x  — 10  sin  X  cos^  x  — 10  cos^  x  + 15  sin^  x  cos  x  +  7  sin  x  cos  x  +  cos  x), 

hence 

/(<>)=  l,/'(0)=  l,/"(0)=  l,/"'(0)=  0,/'v(0)=  -  3,/v(0)=  -  8, 

therefore  e"°*  =  1  +  x  +  —  -  — x*  -  — x^  +  ... 

2!     4!         51 

=  1  +  x  +  -x2-lx*-— x6+  .... 
2  8  15 

There  is  no  observable  law  of  succession  for  the  numerical  coefficients, 
and  the  coefficient  of  x"  is  not  expressible  as  a  simple  function  of  n. 


56-57.]  EXPANSION  OF  FUNCTIONS  93 

57.  Development  of  /(«)  in  powers  of  a;  -  a.  It  was  seen  in 
Art.  56  that  if /(a:)  or  any  of  its  derivatives  be  discontinuous 
in  the  vicinity  of  a;  =  0,  then  /(a;)  has  no  development  in 
powers  of  x. 

It  will  be  shown,  however,  that  if  these  successive  func- 
tions be  continuous  in  the  vicinity  of  some  other  value  x  =  a, 
then /(a;)  will  have  a  development  in  powers  oi  x  —  a,  which 
will  be  a  true  representative  of  the  function  for  values  of  x 
within  a  certain  interval  in  the  vicinity  of  a;  =  a. 

First,  to  find  the  form  of  such  development,  let 

f(x)  =  A  +  B(x  -  a)  +  0(x  -  a)2 

+  I)(x-af  +  E(x-ay+"-  (1) 

be  regarded  as  an  identity,  the  coefficients  A^  B,  C,  ■-•  being 
independent  of  x.  With  the  same  hypothesis  for  the  vicinity 
of  a;  =  a  as  was  before  made  for  the  vicinity  of  a;  =  0,  differen- 
tiation furnishes  the  additional  identities  : 

/(a;)  =  5  +  2(7(a;-a)4-     3i)(a:-a)2+  4^(a;-a)3+- 

f'(x)^        2(7  +3-2i)(a;-a)  +      4  •  3^(a:-a)2+- 

f"(x)=  3-22)  +4.3-2J7(ar-a)  +••• 

If,  now,  the  special  value  a  be  given  to  a:,  the  following 
equations  will  be  obtained  : 

/(a)=^,    fia)=B,    /"(a)  =  2(7,    /'"(«)=  3.22),  ... 
hence, 

^=/(a),     5=/(a),     C=£^,     D=l^,... 

Thus  the  coefficients  in  (1)  are  determined,  and  the  re- 
quired development  is 

fix-)  =f<ia)  +fia)<ix  -  a)  ^^-^ix  -  a^  +-^^(^  -  «)' 

+  ...+.^^(2:  _«)«  +  -.     (2) 


94  DIFFERENTIAL   CALCULUS  [Cii.  IV. 

Ex.  1.    Expand  log  x  in  powers  oi  x  —  a. 

Here  f(x)  =  1...^  x,f'(x)  =  Kf"(x)  =  -  \f"ix)  =  ^-^..., 
X  x^  x° 

hence,     /(«)  =  log  aj'{a)  =  -,/'(")  =  -  -^. ./""(")  =  -^'  ' 

y>.(,)^(-l)"- (»-!)!, 
•^   ^  ^  a" 

and,  by  (2)  the  required  development  is 

logx  =  logrt  -}-  -  (x  -  ^0  -  :r^(-f  -  ay  +  ^-i(-c  -  ^0" 

+  (zd^):^(x-a)»+-.. 
The  condition  for  the  convergence  of  this  series  is  that 


li.n   r(3r-«)"+^  .(^-^)"|i^ii. 


l.«f.,  —  ]<  1, 

X  —  n\<\  a, 

()<j-<2rt. 

It  will  be  shown  presently  that  this  is  also  the  inten'al  of  equivalence 
of  logx  and  its  developinent.  This  series  may  be  called  the  develop- 
ment of  logx  in  the  vicinity  of  r  =  o.  Its  development  in  the  vicinity  of 
j:  =  1  has  the  simpler  form 

log:c  =  X  -  1  -  K-r  -  1)2  +  \{x  -  1)8  -  ..., 

which  holds  for  values  of  x  between  0  and  2. 

Ex.   2.   Show  that  the  development  of  -  in  powers  of  x  —  a  is 


-  —  (-^  -  «)  +  -^  (3-  -  o)-  -  -:  (^  -  «)"  +  — , 

n-  n"  rt* 


1  =  1-1 

x      a      a- 
and  that  the  series  is  convergent  from  x  =  0  to  x  =  2  a. 

Ex.  3.   Develop  e*  in  powers  of  x  —  2. 

Ex.  4.    Develop  x^  —  2 x'^  +  nx  —  7  in  powers  of  r  —  1. 

Ex.  5.   Develop  H  t/'^  —  \i  y  +  7  in  powers  of  ;^  —  3. 


57-60.]  EXPANSION   OF  FUNCTIONS  95 

58.  Remainder.  The  second  restriction  imposed  upon  the 
series  in  order  that  it  may  be  a  correct  representative  of  the 
generating  function,  is  that  the  remainder  after  n  terms  may 
be  made  smaller  than  any  given  number  by  taking  ?*  large 
enough. 

Before  getting  the  general  form  for  this  remainder  it  is 
necessary  to  prove  the  following  lemma. 

59.  Rolle's  theorem.  If  /(a?)  and  its  first  n  +  1  deriva- 
tives are  continuous  for  all  values  of  x  between  a  and  b,  and 
/(*)'/(^)  both  vanish,  then /'(a;)  will  vanish  for  some  value 
of  X  between  a  and  b. 

By  supposition  f(x)  cannot  become  infinite  for  any  value 
of  X,  such  that  a<x<b;  and  ii  f\x)  does  not  vanish,  it 
must  always  be  positive  or  always  be  negative  ;  hence, /(ar) 
must  continually  increase  or  continually  decrease  (Art.  20). 

This  is  impossible,  as/(a)=  0  and/(ft)=  0,  hence  at  some 
point  x  between  a  and  b,  f(x)  must  cease  to  increase  and 
begin  to  decrease,  or  cease  to  decrease  and  begin  to  increase. 

This  point  x  is  defined  by  the  equation /'(i)=  0. 

To  prove  the  same  thing  geometrically,  let  y  =/(a;)  be 
the  equation  of  a  continuous 
curve,  which  crosses  the  a;-axis 
at  distances  x  =  a,  x  =  b  from  the 
origin  ;  then  at  some  point  be- 
tween a  and  b  the  tangent  to  the 
curve  is  parallel  to  the  a:-axis,       ,  ,.     ,„ 

'^  !•  iG.  12. 

since  by  supposition  there  is  no 

discontinuity  in  the  direction  of  the  tangent.     Hence 

60.  Form  of  remainder  in  development  of  /(»)  in  powers 
of  05  -  a.     Let  the  remainder  after  n  terms  be  denoted  by 


96  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

M„(x,  a),  which  is  a  fuiiotion  of  x  and  of  a  as  well  as  of  n. 
From  the  form  of  the  succeeding  terms,  J{„  may  be  con- 
veniently written  in  the  form 

j?„(2;,  a)  =  ^^^^^(f)(x,  a), 

and  then  the  problem  is  to  determine  ^(x,  a),  so  that  the 
following  may  be  an  algebraic  identity  : 

fCx)  =f{a)  +f(a)(x  -  a)  +-'-^  (x-ay+- 

(n  —  1) !  ^  nl  ^  ^  ^ 

in  which  the  right-hand  member  contains  only  the  first  n 
terms  of  the  series,  with  the  remainder  after  n  terms.     Thus 

f(x)-f(a)  -f(aXx  -  a)  -^-^  (x  -  af  -  ••. 

Let  a  new  function  F{z)  be  defined  as  follows : 
F(z)=f(x)-f(z}-fCzXx  -  ^)  --YT-i^  -  'Y  -  - 

_/!:!«(,  _,)»-._i(^(,_.).,  (3) 

(n  —  \)\  n\  ^ 

in  which  the  right-hand  member  is  obtained  from  (2)  by 
replacing  a  by  the  variable  z  in  every  term  except  <j>(x,  a). 

This  function  F{z^  vanishes  when  z  =  x,  by  inspection  ; 
and  it  also  vanishes  when  z  =  a,  by  (2)  ;  hence,  by  Rolle's 
theorem,  its  derivative  F'(z')  vanishes  for  some  value  of  z 
between  x  and  a,  say  z^     But 

F'(z-)=-f(z)+f'(z)-f"(zXx  -  z)+f"(zXx  -  2)  -  ... 

f"(z)     r  N„  1        <^Ca;,  a)    .  .„  , 


60-61.]  EXPANSION   OF  FUNCTIONS  97 

and  these  terms  cancel  each  other  off  in  pairs  except  the  last 
two  ;  hence 

then  since  F'(z)  vanishes  when  z  —  Sj,  it  follows  that 

<^(2:,a)=/«(0i),  (4) 

wherein  z^  lies  between  x  and  a,  and  may  thus  be  represented 
by  z^  =  a-\-  d(x  —  a), 

where  ^  is  a  positive  proper  fraction.     Hence  from  (4) 
<f>(x,a}=f"la  +  e(x-a)'\, 

and  R„ix.  a)=Q^±^--^(x--ar* 

The  complete  form  of  the  expansion  of  /(a?)  is  then 
/(£c)  =  f(a)  +  f\a)  (05  -  a)  +  ^^  (ac  -  a)^  +  ... 

^f^-Ha)  (^  _  „)»-!  +r («  +  e(a;  -  g))  ^^  _  „)n    (5) 
(M-l)I  n! 

in  which  w  is  any  positive  integer.  The  series  may  be  car- 
ried to  any  desired  number  of  terms  by  increasing  w,  and  the 
last  term  in  (5)  gives  the  remainder  (or  error)  after  the  first 
n  terms  of  the  series.  The  symbol  /"(a  +  6(x  —  «))  indi- 
cates that  /(a;)  is  to  be  differentiated  n  times  with  regard  to 
X,  and  that  x  is  then  to  be  replaced  by  a  4-  B^x  —  a). 

6L  Another   expression   for   the   remainder.     Instead   of 
putting  Rn(x,  a}  in  the  form 

(x  —  aY,.       -. 
^ z-^<^(x,  a), 


*  This  form  of  the  remainder  was  found  by  Lagrange  (1736-1813),  who 
published  it  in  the  M^moires  de  I'Academie  des  Sciences  k  Berlin,  1772. 


98  DIFFEHENTIAL   CALCULUS  [Ch.  IV. 

it  is  sometimes  convenient  to  write  it  in  the  form 
Ii„(^x,  a)  =  (x  —  a)ylr(x,  a). 
Proceeding  as  before,  the  expression  for  F'{z^  will  be 

and  this  is  to  vanish  when  z  =  z^;  hence 

in  which     z^  =  a  -\-  6(x  —  a),  x  —  z^={x  —  a)(\  —  0}  ; 
thus     ^{r{x,  ^^^/"(«  +  ^<^^-^»(l-^)»-i(^.  _  a)"-\ 

and       7^„(a^,  a)  =  (l  -  ^)«-i/"(«  +  ^(^  -  «))  .^  _  ^y,  * 

(n  —  1)  I 

An  example  of  the  use  of  this  form  of  remainder  is  fur- 
nished by  the  series  for  log  a;  in  powers  of  x  —  a,  when  x  —  a 
is  negative,  and  also  in  Art.  64. 

Ex.  1.    Find  the  interval  of  equivalence  of  logx  and  its  development 
in  powers  of  x  —  a,  when  a  is  a  positive  number. 
Here,  from  Art.  57,  Ex.  1, 

X" 

hence        /"(a  +  e(x  -  a))\  =  \        (^Zll)!        , 

and  R„(x,a)\  =  \ (\-  ">" =  l\       ^  "  "       1". 

Now  the  interval  of  equivalence  is  not  wider  than  the  interval  of  con- 
vergence; hence,  by  Art.  54,  the  first  condition  of  equivalence  is  that 
X  —  n  be  numerically  less  than  a.     First  let  x  —  a  be  positive,  then  when 


*  This  form  of  the  remainder  was  found  by  Oauchy  (1789-1857),  and  first 
published  in  his  "  Lemons  sur  le  calcnl  infinitesimal,"  1826. 


61-62.]  EXPANSION   OF  FUNCTIONS  99 

it  lies  between  0  and  a  it  is  numerically  less  than  a  +  6(x  —  a),  since  6  is 
a  positive  proper  fraction  ;  hence  when  n  =  oo 

r    /.~"      .\"  =  0,  and  R„(x,  a)=  0. 
La  +  0{x  —  «)  -I 

Again,  when  a:  —  a  is  negative,  and  numerically  less  than  a,  the  second 
form  of  the  remainder  must  be  employed.     As  before, 

hence  Ru(x,  a)\  =  \n  -6)"-^ (^,~  ^^^ 

1  =  1(1  _^)«-i i2^1^ 


|_|r(a  -x)-e(n  -a)l"-'  a-x 


The  factor  within  the  brackets  is  always  less  than  1,  hence  the 
(n  —  l).st  power  can  be  made  less  than  any  given  number,  by  taking  n 
large  enough.     This  is  true  for  all  values  of  x  between  0  and  a. 

Therefore,  logo:  and  its  development  in  powers  oi  x  —  a  are  equivalent 
within  the  interval  of  convergence  of  the  series,  that  is,  for  all  values  of 
X  between  0  and  2  a. 

Ex.  2.  Show  that  the  development  of  ar  2  in  positive  powers  of  a;  —  a 
holds  for  all  values  of  x  that  make  the  series  convergent ;  that  is,  when  x 
lies  between  0  and  2  a. 

62.  Form  of  remainder  in  Maclaurin's  series.  The  above 
form  of  remainder  is  at  once  applicable  to  Maclaurin's  series 
by  putting  a  =  0.     The  result  is 

/(^) = /(O) + /'(0)x +^x^  + ...  +  p::^a.-» +>^^«. 

21  (»  — 1)1  n! 

The  remainder  formula 

n  I 

will  now  be  used  to  show  that  the  interval  of  equivalence  of 
any  one  of  the  ordinary  functions,  and  its  development  in 

DIFF.  CALC.  8 


100  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

powers  of  a;,  is  co-extensive  with  the  interval  of  convergence 
of  the  development  itself. 

The  following  lemma  will  be  useful  in  several  cases. 

63.    Lemma.     When  x  has  any  finite  value,  however  great, 
and  w  is  a  positive  integer,  then 

—  =  0  whenw  =  Qo. 


n 


X'       ,,  T 


r+l 


For,  let  w^  =  — ^,  then  w^+,  =  - — — — , 

r\  (r  +  1)1 

and  !Vti^_^. 

Ur       r  +  l 

Now,  however  large  be  the  assigned  value  of  x,  it  is  possi- 
ble to  take  r  so  great  that 

<  a;, 


r  +  l 

where  k  is  some  proper  fraction,  and  then  for  terms  subse- 
quent to  M^,  the  ratio  of  each  term  to  the  preceding  term 
will  be  less  than  the  fixed  proper  fraction  h  ;  hence,  by  6  (a) 
of  Art.  59,  these  successive  terms  approach  nearer  and  nearer 
to  zero  as  a  limit. 

64.    Remainder  in  the  development  of  «=*,  sin  ac,  cos  £c. 

If  /(a;)=  a^  then  f\x)=  a^loga)",  f\ex)=  a«^(loga)», 

and  RJx)  =  a'Xlog  ay.—  =  a'^S^  ^"^  ^^"  ; 

but  C^^"g^)",i,o,  when  w  =  oo,  by  Art.  63  ;  and  a"^  is  finite, 
n  ! 

when  X  is  any  finite  number,  however  great ;  hence 
R„(x)=  0,  when  w  =  oo. 


62-65.]  EXPANSION   OF  FUNCTIONS  101 

Again  let  f{x)  =  sin  a;,  then  f"{x~)=  sin  (a;  4-^)  by  Art. 
49;  hence  ^         ^^ 

nex)  =  sin(^^a:  +  fy  and  R„{x)  =  sm(^Ox  +  y) -f^, 

but  sin  f  ^a;  +  —  J  never  exceeds  unity  for  any  value  of  x  or 
of  w,  hence,  by  Art.  63,  J?„(a;)=0,  when  w=  oo. 

Similarly,  if  /(a;)  =  cos  a:,  /"(a;)  =  cos  ( a;  +  ^  j,  and  as 
before,  -K„(a:)=0,  when  n  =  cc.  ^  ^ 

Hence  the  developments  of  a^,  sin  a;,  cos  a;,  hold  for  every 
finite  value  of  x. 

Ex.  1.   If /(x)  =  sin  X,  compute  -^3(3:),  when  a:  =  |  tt  radians. 

Ex.  2.  Expand  sin  ax  by  Maclaurin's  theorem,  and  determine  the 
remainder  after  7  terms,  counting  the  terms  that  have  zero  coefficients. 

Ex.  3.  Show  that  the  absolute  error  in  stopping  the  series  for  sin  x, 
cos  ar,  at  any  term,  is  less  than  the  next  term  of  the  series. 

Ex.  4.  Show  that  the  relative  error  in  stopping  the  series  for  e^,  at  any 
term,  is  less  than  the  next  term  of  the  series;  the  relative  error  being  the 
ratio  of  the  absolute  error  to  the  true  value  of  the  function  to  be  computed. 

Ex.  5.   Prove  by  expansion  that 

e>A=Tx  ^  gv^x  _  2  cos  a:,  e^-^^'  -  e""^^'^  =  2V-  1  sin  a:; 

and,  hence  by  addition,  e''^-^'  =  cos  x  +  V  —  1  sin  x. 

Ex.  6.   From  the  last  example,  prove  De  Moivre's  theorem : 

(cos  X  +  V—  1  sin  x)™  =  cos  mx  +  V—  1  sin  mx. 

65.  Taylor's  series.*  It  will  next  be  shown  how  to  write 
down  the  development  for  a  function  of  the  sum  of  two 

*So  named  from  its  discoverer.  Dr.  Brook  Taylor  (1685-1731),  who  pub- 
lished it  in  his  "  Methodus  Incrementorum,"  1715  ;  but  the  formula  remamed 
almost  unnoticed  until  Lagrange  completed  it  by  finding  an  expression  for 
the  remainder  after  n  terms  (Art.  60).  Since  then  it  has  been  regarded  as 
the  most  important  formula  in  the  Calculus. 


102  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

numbers  in  ascending  powers  of  either  number,  and  also  an 
expression  for  the  remainder  after  n  terms  of  the  series. 
If  in  the  identity  of  Art.  60,  which  gives  an  expression 
for  f{x)  in  powers  of  x  —  a,  the  letter  x  be  everywhere 
replaced  by  a;  +  a,  then  x  —  a  will  be  replaced  by  a;,  and  the 
identity  will  assume  the  form 

,    f-^{a)    „-i  ,  f"  (a  +  dx)    „         .1. 
+  7 7-7  x"  1  +  -^  , x^,       (1) 

(w— 1);  71. 

in  which  x,  a  are  any  two  numbers,  n  is  any  positive  integer, 
and  6  is  some  positive  proper  fraction,  which  may  not,  how- 
ever, be  independent  of  the  values  of  the  other  letters. 

If  the  second  form  ot  remainder  be  used,  the  last  term  on 
the  right  will  be  replaced  by  (1  -  ey-^^J^  't^f^x". 

In  the  identity  (1)  the  letters  x  and  a  may  be  interchanged, 
hence  the  expansion  for /(a;  +  a)  in  powers  of  a  is 

f(x  +  a)  =f(x)  +f(x)a  +-^^a2  +  ... 

+  /!:^<,^.+/!££±Ma.;       (2) 
(n—l)\  n\ 

and  the  second  form  of  remainder  is 

R^ix,  a)  =  (1  -  ey-'  -^yty  a\  (3) 

(n-\y. 

Ex.  1.    Expand  (a  +  2?)"*  in  ascending  powers  of  a;,  and  finH 
the  interval  of  equivalence. 

Here  /(a  +  a:)  =  (a  -|-  «)"», 

hence  /(^)  =  ^t 


66.]  EXPANSION  OF  FUNCTIONS  103 

and     f'(x)  =  mx'"-\  f" {x)  =  m(w  - 1  )x'"--, •  • ., 

/»(a;)  =  m(m  —  V)'-'(m  —  n  +  l  )x"'-'' , 
f(^a}=a"',  f\a)  =  ma"'-\  f" (a)=m{m-\)a"'-'\"', 
f''(a)  =  m{m-l)  •••  (w-m  +  I  )«'"-", 

therefore     (a  +  a:)"*  =  a'"  +  ma'"-ia;  Tf  ^^^7^^  a"'-^3y^+  •- 

(w-1)! 
in  which,  from  the  first  form  of  remainder, 

^m(m-l)-(m-»  +  l)  g       /_^_Y 

w  !  \a  +  ^a;/ 

It  will  first  be  shown  that  the  factor 

TwCw  — 1)  •••  Cwi  — w  +  1)   ,  , 

—A i — 1 ! — ^  :^oo  when  n  =ao  ; 

nl 
for,  if  it  be  denoted  by  m„,  then 

w„       w  + 1 

and  this  ratio  can,  by  taking  n  large  enough,  be  made  as  near 
unity  as  may  be  desired;  but  it  can  never  exceed  unity,  hence 
the  successive  values  of  m„  will  approach  the  limit  zero  or  a 
finite  number,  when  n  =  oo   (Art.  54,  6  (a}). 
Next,  the  expression 

I )  =0  when  w=qo, 

\a  +  6xj 

if  X  be  positive  and  less  than  a.     Hence 

M„(a,  x)  =  0,  when  n=oo, 

if  X  be  positive  and  less  than  a. 


104  DlFFt^RENTlAL  CALCULUS  [Ch.  IV. 

Since  the  interval  of  convergence  is,  by  Art.  54,  from 
x=  —a  to  x  =  a,  it  remains  to  examine  the  value  of  Ii„{a,  x) 
when  X  is  negative  and  numerically  less  than  a.  For  this 
purpose  it  is  necessary  to  use  the  second  form  of  remainder, 

R„(a,x}  =  (l  —  dy  ^—^ ^ ^— i ■ — ^(a  +  dx)"'  "x" 

m(m-l)  -  (m-n  +  1)    (a  +  0xy--^fl--d_ 

+  i 

But  when  -  is  negative  and  less  than  1,  the  expression 
a 

-^ g 

is  a  proper  fraction,  hence  its  (w  — l)st  power  ap- 


(n-1)!  •         a-'        \  +  0^)       w' 

a 


1  +  0- 


a 
proaches  zero  as  a  limit ;  and  it  can  be  shown  as  before  that 

the  factorial  expression  is  not  infinite.     Hence 

^„(a,  a;)  =  0,  when  w=cx), 

if  X  lies  between  —  a  and  +  a. 

This  is  therefore  the  required  interval  of  equivalence. 

Ex.  2.    Expand  log  (x  +  a)  in  powers  of  a;,  and  find  the 
interval  of  equivalence. 

Here  f(x-{.a)=  log  (x  +  a), 

f(x)=\ogx, 

X  or  a^ 

X^ 

hence 

log(a:  +  «)=  loga +--^a:2  + -igS^s  ... 

(w-l;a"-i  n(^a  +  dxy 


66.]  EXPANSION   OF  FUNCTIONS  ^  105 

This  expansion  could  also  be  obtained  from  the  develop- 
ment of  log  X  in  powers  of  x  —  a,  in  Art.  57. 
Similarly, 

log(a-a^)=loga-^-— -ar^-— -2^-..  -.        ^" 


a      2a2  3a3  (n-l^a"-' 


n(a  —  6^xy 
When  a  =  1,  these  series  become 

\ogii-rx)=x --  +  ---+  •"+^ ^— 

ii  O  4  71  —  1 

(-1)"-V^ 
w  (1  +  ^:c)"' 

/*«2       /y5        /*4  /v.n — 1  ^» 

\og(l-x^=-x------ 


2       3       4  w-1      n{\-d^xY 

in  which,  as  in  Ex.  1  of  Art.  61,  the  remainder  R„(x)=(i^ 
when  w  =  30,  if  —  1<  a:  <  1.       Also,  by  subtraction, 

which  can  be  used  for  computation  when  x  is  numerically 
less  than  unity. 

This  identity  can  be  thrown  into  a  form  suitable  for  the 
numerical   calculation   of   the    logarithm    of    any   number ; 

for,  put 

1  +  a^^n  +  h      ^^^^     ^^       h       . 


1  —  X         n  2n  -\-  h* 

from  which 

^^\    n    )         \2n  +  h^{2n  +  hy      b(2n  +  hy  j 

This  is  an  identity  for  all  positive  values  of  n  and  A,  since 
the  original  condition  2:|<|1  is  replaced  by |<|  1, 


106  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

and  the  latter  condition  is  always  fulfilled  when  n  and  h 
are  positive. 

Suppose  it  is  required  to  find  log  10.  This  could  be 
done  by  putting  w  =  1,  A  =  9  in  the  last  equation,  but  the 
series  thus  obtained  would  converge  too  slowly  to  be  of 
practical  value.  Let  log  2  be  first  calculated  by  putting 
both  n  and  h  equal  to  1 ;  thus 

^  [3^3     38^5      36^7      37^      J 

Next,  put  w  =  8,  A  =  2, 

W  10  =  3 loff  2  +  -f-  + 1  •  -  +  -  •  —  +  -  •  —  +  -I- 

The  numerical  work  can  be  greatly  facilitated  by  proper 
arrangement  of  terms.  The  result  correct  to  8  places  of 
decimals  is  log  10  =  2.30258509. 

The  student  should  bear  in  mind  the  distinction  between 
theoretical  and  practical  convergence.  Here,  only  theoreti- 
cal convergence  has  been  considered.  To  make  a  series 
practically  useful,  i2„  should  be  so  small  that  after  ten  or 
twelve  terms  it  could  be  neglected  without  affecting  the 
desired  numerical  approximation.  Sometimes,  however,  the 
expression  for  _B„  does  not  lend  itself  easily  to  a  numerical 
estimate  of  the  error  made  in  stopping  the  series  at  a  given 
term.  The  method  of  comparison  with  a  descending  geo- 
metrical progression,  stated  in  (6)  of  Art.  54,  and  illustrated 
in  Ex.  1  of  Art.  56,  and  in  Exs.  1,  2,  3  of  Art.  67,  will  be 
found  very  useful  in  practice. 

Ex.  3.  Expand  sin  (x  -\-  y)  in  ascending  powers  of  y.  Hence  verify 
that  sin  (x  +  y')=  sin  x  cos  y  +  cos  x  sin  y. 


65-66.] 


EXPANSION  OF  FUNCTIONS 


107 


Increment  of  function   in 
An  important  special  case 


66.   Theorem   of   mean   value, 
terms  of  increment  of  variable. 

of  Taylor's  theorem  is 

f(x  +  K)  =^f(x)  +  hf(x  +  eh\  (1) 

which  is  obtained  by  putting  n  =  1,  in  equation  (2)  of  Art. 
65,  and  replacing  a  by  h. 

If  fix)  be  transposed,  and  Aa;  be  written  for  A,  the  identity 
may  be  written 

Lf(x)  =  ^x  'f'(x  +  e'^x\  (2) 

which  expresses  that :  The  increment  of  the  function  is  equal 
to  the  increment  of  the  variable  multiplied  by  the  value  of 
the  derivative  taken  at  some  intermediate  value  of  x. 

This  theorem  is  true  whether  the  increments  be  large  or 
small.  It  has  a  simple  geometrical  interpretation.  Since 
f{x)  is  continuous,  it  can  be  represented  by  a  curve  whose 
equation  is  y  =f(x). 

In  Fig.  13,  let 

x  =  ON,  x  +  ^x  =  OR, 
f(x)  =  NH,  fix  +  Aa;)  =  BK, 
then       £^f(x)  =  MK,  and 

Ax  MM 

hence,  /'(^x  +  O-  Ax)  =  tan MffK. 

But  f'(x  +  d'  Ax)  is  the  slope  of  the  tangent  at  some 
point  S  between  H  and  K',  thus  the  theorem  of  mean  value 
expresses  that  at  some  point  between  H  and  K  the  tangent 
to  the  curve  is  parallel  to  the  secant  HK.  This  is  self- 
evident,  geometrically ;  and  has  already  been  mentioned  in 
Art.  59. 


108  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

Ex.  1.   Verify  the  theorem  of  mean  value  for  the  function  /(j")  =  x^. 

Here  /(x  +  A)  =  (x  +  h)'^  =  a;2  +  A  •  2  (a;  +  Oh), 

which  is  evidently  a  true  identity  when  6  =  \.     In  most  cases  the  exact 
numerical  value  of  the  proper  fraction  6  is  not  so  apparent. 

When  the  given  increment  of  x  is  small  and  the  increment 
of  the  function  is  desired,  it  is  sometimes  sufficiently  accurate 
in  practical  computation  to  replace  f'(x  +  0-  Ax)  in  equa- 
tion (2)  by  its  approximate  value  f'(x)^ 

then  Af(x)  =  Ax  -f(x),  (3) 

in  which  the  error  is,  by  Taylor's  theorem, 

^(iAxy-f"(x  +  e-Ax\ 

a  term  of  the  second  order  of  smallness. 

A  second  approximation  to  the  value  of  A/ (a;)  is  given  by 

Af(x~)  =  Ax  'f(x)  +  1  {Axy  ■fix'),  (4) 

in  which  the  error  is   I  {Ax^  '/'"(x  +  6  •  Ax),  of  the  third 
order. 

The  third  approximation  is  obtained  by  adding  the  term 
^(^Ax)^f"'(x),  and  the  error  will  tlien  be 

^(iAxy-f^(x  +  d'Ax). 

Ex.  2.  Compute  the  first,  second,  and  third  approximations  to  the 
increment  of  log  x  when  x  changes  from  10  to  10.1. 

Ex.  3.  Show  how  to  compute  the  difference  for  one  minute  in  a  table 
of  natural  sines. 

Increment  of  the  increment.  Let  y  =f(x)  be  a  function 
which  can  be  developed  in  the  vicinity  of  x  —  x^;  and  let  x 
have  the  three  successive  equidistant 
^'V  values  x^  —  A,  x^,  x-^  +  h.  When  x 
changes  from  x^  —  h  to  x^,  let  y  take 
the  increment  A^y  =f(x{)  —f(x^  —  h) 
=  h-f'  {x^)  -  1  h^f"  (x^  -  eh)  ;     and 


66-67.]  tJ^PANStON  Of  FUNCTIONS  109 

when  X  clianges  further  from  a:^  to  z^  +  /i,  let  1/  take  the 
increment 

let  the  difference  of  these  successive  increments  of  y  be 
written  A  (Ai/)  or  A^^  ; 

then     A2^  =  A2^-Aiy=|[/"(a:i  +  ^'A)+/"(x,-^A)].       (5) 

'  This  result  may  be  expressed  in  words  thus :  The  incre- 
ment of  the  increment  of  the  function,  corresponding  to 
successive  equal  increments  of  the  variable,  is  equal  to  the 
square  of  the  latter  increment  multiplied  by  half  the  sum  of 
the  values  of  the  second  derivative  taken  at  intermediate 
values  of  the  variable  on  each  side  of  its  middle  value.  This 
may  be  called  the  theorem  of  mean  value  for  the  second 
derivative. 

Ex.  4.    Prove  that  A^y  is  an  infinitesimal  of  the  same  order  as  (Ax)^. 

Ex.  5.  Show  how  to  compute  the  change  in  the  ditference  for  one 
minute  in  exercise  3. 

Limit  of  the  ratio  of  t^y  to  (Aa:)^.  In  equation  (5), 
replace  h  by  A2:,  divide  by  (Ax)'^,  and  take  the  limit  of  both 
members  as  Aa;  =  0,  then 

lim     A^  _  ^ 

67.  To  find  the  development  of  a  function  when  that  of  its 
derivative  is  known.  Development  of  the  anti-trigonometric 
functions. 

The  derivative  of  an  anti-trigonometric  function  being  an 
algebraic  binomial,  it  is  easy  to  expand  it  by  the  binomial 
theorem  ;  it  is  now  proposed  to  show  how  to  use  the  develop- 
ment of  the  derivative  to  determine  the  coefficients  in  the 


110  DIFFERENTIAL   CALCULUS  [C  i.  IV. 

development  of  the  given  function,  so  as  to  avoid  the  labor 
of  successive  differentiation. 

1.    Power-series  for  t'Ar\~^x. 

Assume,  within  an  interval  including  x  =  0,  the  identity 

t2ixr^x  =  A-\-Bx-\- Cji^  +  J)3^  +  E2^  +  '-.  (1) 

With   the   same   preliminary  hypothesis   as' in  Art.    56, 
differentiation  furnishes  the  identity 

1 


1+aP' 


=  J5  +  2  Cx  -f-  3  i)a^  +  4  .K»8  +  ...,  (2) 


but,  within  the  interval  from  —  1  to  +  1,  the  left  member  is 

identical  with 

1  -  «2  +  cc*  -  a:«  +  ..-, 

hence,  within  a  certain  interval  including  a;  =  0,  there  exists 
the  identity 

\-x'^-\-7^-3^+"'=B+2  Cx-{-^I)x^+4:Ej(?-\-"'.      (3) 

therefore  ^  =  1,   (7=  0,  D  =  -  ^,  .£7  =  0,  ^  =  -J,  .... 

The  first  coefficient  A  is  found  to  be  zero  by  putting 
a;  =  0  in  (1),  hence 

t2in~^x  =  X —\x^ +  \a^ —  \x'' +  •".  (4) 

The  interval  of  convergence  of  this  series,  found  by  the 
usual  method,  is  from  a;  =  —  ltoa:=l. 

To  show  that  this  is  also  the  interval  of  equivalence  of  the 
function  and  the  series,  and  thus  to  establish  the  validity  of 
the  development,  let  Rn(x)  denote  the  remainder  obtained 
by  subtracting  the  sum  of  the  first  n  terms  from  the  func- 
tion, then 

tan-la;  =  x  -  \7^+\a^  -  ...  ^:  -^^  +  R,(x\       (5) 


67.]  EXPANSION   OF  FUNCTIONS  111 

hence  by  differentiation  with  regard  to  x, 

-i—  =l-x'  +  x*-.-'T  a:2"-2  +  R„'(x},  (6) 

1  4-  ar 

therefore  RJ(x)  is  the  remainder  after  n  terms  obtained  by 
dividing  1  by  1  -f  a::^, 

Rnix)=±^.  (7) 

By  the  theorem  of  mean  value,  Art.  66, 

i2„(x)  =  /?„(0)  +  xRJiOx),  [0  <  ^  <  1 

but,  from  (5),  ^„(0)  =  0  ;  and  when  x  is  less  than  1,  Ox  is 
less  than  1,  hence,  by  (7), 

i2„'(^a;)=  0,  when  w  =  oo  ; 
therefore  „^V"^  R„(x^  =  0. 

Thus  the  interval  of  equivalence  is  from  —  1  to  +1. 
Ex.  1.  Compute  tan-^  J,  tan-*  J,  tan  *  1 ;  and  hence  the  value  of  tr. 

tan-4  =  i  -  Ki)' +  iG)*  -  KiP+ - 
=  +  .5 

-  .0416667 
+  .0062500 
-.0011162 
+  .0002170 

-  .0000444 
+  .0000095 

-  .0000023 
+  .0000005 

-  .0000001 


.4636473  +  radians. 

tan-H=J-ia)'  +  ia)^  -• 
=  +  .3333333 

-  .0123457 
+  .0008230 

-  .0000653 
+  .0000056 

-  .0000005 
.3217506+  radians. 


112  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

To  find  tan-'  1,  use  the  formula: 

tan-i-  +  tan-i-  =  tan-i  -i±i_  =  tan->  1  =  -;  ' 

hence  -  =  .4;):56473  +  .3217506  =  .7853979  +  .••, 

4 

and  ir  =  3.1415916  ••. 

In  the  first  series,  to  estimate  roughly  the  error  made  by  stopping  at 
the  tenth  term,  it  may  be  observed  that  the  ratio  of  any  term  to  the  pre- 
ceding is  numerically  less  than  \,  and  approaches  ^  as  a  limit ;  hence,  if 
all  the  terms  after  Mj^  were  positive,  their  sum  would  be  less  than  the 
geometric  series 

"io(l  +  i  +  i +■•••), 

which  is  less  than  Mj„;  moreover,  since  the  alternate  terms  are  negative, 
it  follows  that  the  error  made  in  stopping  at  Hj„  is  really  nmch  less' than 
UjQ,  and  is  thus  less  than  one  unit  in  the  seventh  decimal  place. 
Similarly,  in  the  second  series,  the  error  is  much  less  than 

«6a  +  i  +  ^v +  •••). 

i.e.,  less  than  -^,  or  less  than  2J  units  in  the  seventh  place.    Thus  the 

error  in  the  value  of  —  is  less  than  3^  such  units,  and  the  error  in  the 

4 
value  of  TT  is  less  than  \\  units  in  the  sixth  place. 

Therefore  the  numerical  value  of  tt  lies  between  3.1415916  and 
3.1415931. 

2.    Power-series  for  sin~^a;. 

Proceed  as  before,  and  use  the  development 

(l-:r2)-^  =  l  +  i:r2  +  l.|^  +  i.3.|.^  +  ...,  (1) 

then    sm^.=  .-f-.-  +  -.-.-4-^-4.e.y-f-...,     (2) 

in  which  the  interval  of  convergence  is  from  —  1  to  1. 

Let  Rn(x)  be  the  remainder  after  n  terms  in  (2),  then  by 
differentiation,  R„'(x)  is  the  remainder  after  n  terras  in  (1), 

but  Jt„(x)  =  Il„(0)+xRJ{€x)  =  zR„'<iex),     [0<e<l 


67.]  EJiPANSION  OF  FUNCTIONS  113 

and,  by  Art.  65,  Ex.  1, 

R\(jex)  =  0  when  w  =  oo,  if  a;  |<|  1 ;  hence  Rjjc)  =  0  ; 
and  the  interval  of  equivalence  in  (2)  is  from  —  1  to  1. 

Ex.  2.    Compute  sin-i(|),  ^nd  hence  obtain  the  numerical  value  of  ir. 

I  =  sin-Hi)  =  i  +  I  •  Ui)' +  M  •  i  (0' +  M  •  f  •  Ki)' + - 

=  .5000000000 
+  .0208333333 
+  .0023437500  . 
+  .0003487723 
+  .0000593397 
+  .0000109239 
+  .0000021183 
+  .0000004262 
+  .0000000881 
+  .0000000186 

.5235987704  ;    hence  tt  =  3.1415926224+. 

Here  each  term  may  be  used  to  obtain  the  next  by  applying  as  a 
factor  the  corresponding  term  in  the  series  of  ratios : 

1  .1  3-3  5-5  7-7  9.9  11-11 

2.3-4'   4.5-4'    6-7-4'    8-9-4'    10-11-4'    12-13-4' 

To  determine  the  maximum  error  made  by  stopping  at  the  tenth 
term,  it  is  evident  that  the  ratio  of  each  term  to  the  preceding  is  less 
than  \,  and  appi'oaches  ^  as  a  limit ;  therefore  the  sum  of  the  remaining 
terms  is  less  than 

Wio(i  +  A  +  ^+-). 
that  is,  less  than  \  M,g.     Hence  the  error  in  the  value  of  ^  Tr  is  less  than  63 
units  in  the  tenth  place,  and  the  error  in  the  above  value  of  it  is  less  than 
378  units  in  the  tenth  place.     Thus  the  numerical  value  of  tt  lies  between 
3.1415926224  and  3.1415926602.* 

Ex.  3.  Show  that  the  error  made  by  stopping  at  any  term  in  the  series 
for  log  10,  Ex.  2,  Art.  65,  is  less  than  ^  of  the  last  term  used. 

*  Both  of  these  formulas  for  v  were  found  by  Montferier.  The  correct 
value  to  ten  places  is  3.1415926536.  By  various  methods  mathematicians 
have  carried  the  approximation  to  a  much  larger  number  of  places.  Mr. 
Shanks,  of  Durham,  England,  published  the  value  of  -k  to  607  places  in  1853. 
No  other  constant  has  so  much  engaged  the  attention  of  mathematicians. 
See  "Famous  Problems  in  Elementary  Geometry,"  by  Professor  Klein, 
translated  by  Professors  Beman  and  Smith,  1897. 


114  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

EXERCISES 
Derive  the  following  expansions : 

1.  secx  =  l  +  |;  +  ^  +  |ix«  +  i?. 

/yJi  ')*4  i^D 

2.  logsec.  =  |  +  g  +  f^+/e. 

93  7.4         O6  r6 

3.  cos^a;  =  l-a:2  +  ±-f-^  +  i2. 

4!  b! 

4.  e==  cos  a;  =  1  +  X — -—  +  R. 

3!         4! 

6.  log  (1  +  sin  x)  =  a:  -  I  +  1^  -  —  +  i2. 

7.  sin2  X  =  x2  -  —  +  -^^^^  +  R. 

3       32 .  5 


8.  VI  +4x+  12x2  =l+2x  +  4x2  +  i?. 

9.  cos  (x  +  a)  =  cos  x  —  a  sin  x  —  —  cos  x  +  —  sin  x  +  /? 


3! 

cs_  _    , 
2  3  sin^x 


10.  log  sin  (x  +  a)  =  log  sin  x  +  a  cot  x  —  —  csc2  x  +  -; — -, H  i?. 

2  x^ 

11.  e'  sec  X  =  1  +  X  +  x2  H — ; 1-  R. 

o 


12.  log(l+.=^)  =  log2  +  |  +  |'-3^+i2. 

13.  cot-^x  =  Jtt -X  +  ^x3  -  Jx5  +  ....  [a:;<|l 

14.  cot-'x  =  i---i-  +  -^- .  [x|>ll 

X      3  x^      ox* 

15.  tan-ix  =  ^--  +  -i---^+-..  [x|>ll 

2         X        3x3        5J.5  L     I      I 

16.  c.sc-»  X  =  sin-i  -=-  +  o'5-l  +  o'7'^^~^+  '"• 

X      X      2     3x8      24ox° 

17.  Expand  cos~^x  in  powers  of  x;  sec~'  x  in  powers  of  x~^. 


CHAPTER   V 
INDETERMINATE  FORMS 

68.  Hitherto  the  values  of  a  given  function  f(x),  corre- 
sponding to  assigned  values  of  the  variable  x,  have  been 
obtained  by  direct  substitution.  The  function  may,-  how- 
ever, involve  the  variable  in  such  a  way  that  for  certain 
values  of  the  latter  the  corresponding  values  of  the  function 
cannot  be  found  by  mere  substitution. 

ITor  example,  the  function 


sin  a; 


for  the  value  x  =  0,  assumes  the  form  -,  and  the  correspond- 
ing value  of  the  function  is  thus  not  directly  determined. 
In  such  a  case  the  expression  for  the  function  is  said  to 
assume  an  indeterminate  form  for  the  assigned  value  of  the 
variable. 

The  example  just  given  illustrates  the  indeterminateness 
of  most  frequent  occurrence ;  namely,  that  in  which  the 
given  function  is  the  quotient  of  two  other  functions  that 
vanish  for  the  same  value  of  the  variable. 

Thus  if  /(a;)  =  f-^, 

and  if,  when  x  takes  the  special  value  a,  the  functions  <f>  (a;) 
and  -^  (a;)  both  vanish,  then 

DIFF.   CALC. — 9  115 


116  DIFFERENTIAL   CALCULUS  [Ch.  V. 

is  indeterminate  in  form,  and  cannot  be  rendered  determi- 
nate without  further  transformation. 

69.  Indeterminate  forms  may  have  determinate  values.  A 
case  has  already  been  seen  (Art.  16)  in  which  an  expression 
that  assumes  the  form  -  for  a  certain  value  of  its  variable 

takes  a  definite  value,  dependent  upon  the  law  of  variation 
of  the  function  in  the  vicinity  of  the  assigned  value  of  the 
variable. 

As  another  example,  consider  the  function 

x^  —  a^ 

y  = 

X  —  a 

If  this  relation  between  x  and  y  be  written  in  the  forms 

y(x  —  a)=  x^  —  a^,         (x  —  a)Qy  —  x  —  a)  =  0^ 

it  will  be  seen  that  it  can  be  represented  graphically,  as  in 
the  figure  (Fig.  14),  by  the  pair  of  lines 

a;  —  a  =  0, 

y  ~  X  —  a  =  Q. 

Hence  when  x  has  the  value  a  there 

is  an  indefinite  number  of  corresponding 

points  on  the  locus,  all  situated  on  the 

^'*'-  ^'^-  line  X  =  a;  and  thus  for  this  value  of  x 

the  function  y  may  have  any  value  whatever,  and  is  then 

indeterminate. 

When  X  has  any  value  different  from  a,  the  corresponding 
value  of  y  is  determined  from  the  equation  y  =  x  +  a.  Now, 
of  the  infinite  number  of  different  values  of  y  corresponding 
to  X  —  a^  there  is  one  particular  value  AP  which  is  con- 
tinuous with  the  series  of  values  taken  by  y  wlien  x  takes 
successive  values  in  the   vicinity  of  x  =  a.     This  may  be 


68-69.]  INDETERMINATE  FORMS  117 

called  the  determinate  or  singular  value  of  t/  when  x  =  a. 
It  is  obtained  by  putting  x  =  a  in  the  equation  1/  =  x  +  a, 
and  is  therefore  1/  =  2  a. 

This  result  may  be  stated  without  a  locus  as  follows  : 
When  a:  =  a,  the  function 

a^  —  a^ 
X  —  a 

is  indeterminate,  and  has  an  infinite  number  of  different 
values ;  but  among  these  values  there  is  one  determinate 
value  which  is  continuous  with  the  series  of  values  taken  by 
the  function  as  x  increases  through  the  value  a  ;  this  deter- 
minate or  singular  value  may  then  be  defined  by 

lim  ^  —  c? 
*-«  x-a' 

In  evaluating  this  limit  the  infinitesimal  factor  x  —  a  may  be 
removed  from  numerator  and  denominator,  since  this  factor 
is  not  zero,  while  x  is  different  from  a ;  hence  the  determi- 
nate value  of  the  function  is 

lim   ^±«^2a. 
x  =  a       J 

Ex.  1.   Find  the  determinate  value,  when  x  =  1,  of  the  function 
zf  _  1  +  (ar  _  1)1 


which  at  the  limit  takes  the  indeterminate  form  -. 


(a;2  _  1)1  _  a:  +  1 

rm 
0 


This  expression  may  be  written  in  the  form 

(j^-  l)(3:  +  xi+  l)  +  (x^-  \)\(j^  ^  1)1 

(xi  -  l)t(x2  +  1)1  (x  +  1)2  -(xi  -  l)(xi  +  1)' 

from  which  the  infinitesimal  factor  x2  —  1  may  be  removed,  giving 

X  +  x^  +  1  +  (x^  -  1)^  (x^  +  1)1 

(xi  -  l)i  (xi  +  1  )t  (x  +  1)^  -  (x^  +  1) 

which,  when  x  =  1,  approaches  th"  determinate  value  —  \. 


118  BIFFERENTIAL    CALCULUS  [Ch.  V. 

Ex.  2.   Find  the  determinate  value,  when  x  =  a,  of  the  expression 

Vx  —  ^/a  +  y/x  —  a 


Vx8  —  a^  +  Vx*  —  cfix 


by  removing  the  infinitesimal  factor  V  Vx  —  Va. 

70.  Evaluation  by  transformation  and  removal  of  common 
factor.  Sometimes  a  traiisfonnation  must  be  made,  before 
the  common  vanishing  factor  can  be  discovered  and  removed. 

For  instance,  to  evaluate,  when  a;  =  0,  the  expression 


a  —  Va^  —  a^ 


which  takes  the  form   -.      On  multiplying  numerator  and 


denominator  by  a  +  Va^  —  x^^  tlie  fraction  becomes 


which,  by  the  removal  of  the  common  vanishing  factor  aP 
reduces  to 


a  +  Va^  —  otP' 
and  has  therefore,  when  x  is  replaced  by  zero,  the  determi- 
nate value  - — 
"la 

Ex.  1.   Evaluate,  when  x  =0,  the  function 


l-^/^ 


VI  +  X  -  VI  +  ar^ 
[Multiply  numerator  and  denominator  by 

(1  +  vr^T^x  VTT^ + vrr^).] 

Ex.  2.   Evaluate,  when  x=  1,  the  function 

1-xi 
l-(V2x^^^2)i 


70-71.]  INDETERMINATE  FORMS  119 

71.  Evaluation  by  development.  In  some  cases  the  com- 
mon vanishing  factor  can  be  best  removed  after  expansion 
in  series. 

Ex.  1.    Consider  the  function  mentioned  in  Art.  68, 

e^  —  e~^ 
sin  X 

On  developing  numerator  and  denominator  in  powers  of 
X,  it  becomes 


X 

3! 

... 

2a;  4- 

2 
3! 

a?  +  ■" 

2 

x^ 

X  - 

"3! 

.+  ... 

1 

x^ 

which  has  the  determinate  value  2,  wlien  x  takes  the  value 
zero. 

Ex.  2.    As  another  example,   evaluate,  when  a;  =  0,  the 

function 

X  —  sin~^ 
—  sin^a; 
By  development  it  becomes 

/    ^1    a:3  \      ^ 


Removing  the  common  factor,  and  then  putting  a;  =  0,  the 
result  is  \. 

In  these  two  examples  the  assigned  value  of  x,  for  which 
the  indeterminateness  occurs,  is  zero,  and  the  developments 


120  DiFf'ERENTlAL  CALCtTLtTB  [Ch.  V. 

are  made  in  powers  of  z.  If  the  assigned  value  of  x  be  some 
other  number  a,  then  the  development  should  be  made  in 
powers  of  a;  —  a. 

Ex.  3.    Evaluate,  when  a;  =  a,  the  expression 

log  X  —  log  a 
tan  {x  —  a^ 

which  then  takes  the  form  — 

0 

Developing  log  x  in  powers  of  x  —  a  hy  Art.  57  and 
tan  (x  —  a)  in  powers  of  a;  —  a  by  Art.  56,  the  expression 
becomes 

(x  —  a  —  ^(x  —  aYci'^  +  ••• 
X  —  a  -\-  ^(a:  —  a)^  +  ••• 

which,  on  removing  the  common  vanishing  factor  a;  —  a,  is 

1  -  U^  -  a^g-^ -\- -' 
l  +  l(a:-a)2+...  ' 

and  reduces  to  unity,  when  x  takes  the  assigned  value  a. 

In  such  a  case  it  is  usually  convenient  to  write  for  x  —  a 
a  single  letter  h,  and  then  x  is  replaced  by  a  +  h. 

Ex.  4.    Evaluate,  when  a;  =  1,  the  function, 
1  —  x+  log  a; 

1  -  V2¥^^' 

Let  x  —  l  =  h,x=l-{-h,  then  by  developing  in  powers  of 
h,  the  expression  becomes 


which,  on  removing  the  common  vanishing  factor  P,  and  then 
putting  a;  =  1  (that  is,  A  =  0),  reduces  to  the  value  1. 


71-72.]  INDETERMINATE  FORMS  121 


TT 


Ex.  5.    Evaluate,  when  x  =  —,  the  function 


2 
cos  a: 


1  —  sin  a: 


Putting  X  —  —  =  h^  X  =  ^  +  h,  the  expression  becomes 


cos 


V2        /  —  sm  A  6  b 


-         .    fir  ,   j\~l- cosh        h^      A*  h       h^  ^ 

l_sm(^-4-Aj  2-24+-      -2-2i+- 

which  =  T  00  according  as  A=0  from  the  positive  or  negative 

side,  hence 

lim         cos  X  ., 


x  =  \7r^  _ 


Sin  a; 


72.  Evaluation  by  differentiation.     Let  the  given  function 

be  of  the  form  -^  ^^-  ,  and  suppose  that/(a)  =  0,  ^(a)  =  0.    It 
<^(a:) 

is  required  to  find    \"^  /  v  )  ^ 
^  ^  =  «  4>(^x) 

As  before,  let  /(a;),  ^(a;)  be  developed  in  the  vicinity  of 

»  =  a,  by  expanding  them  in  powers  of  a:  —  a,  then 

.,  ,      /(«)  +/'(«)(.  -  a)  +  r(^+f(^-^)\.  _  ay 

fC^)^ rj 

<^(^)      ^(a)+<^'(aX^-a)  +  ^^^i^±|ip^=i^\a;-a)2 

^/^  N^          N  .  f'(a  +  0(x  —  a^').        ^>,o 
/Ya)(a;-q)  +  '^    ^        ,/, I2(x-ay 

"  0^(a)(a;  -  a)  +  ^'^^  +  ^^[^  "  ^->l(a:  -  a)^ 


122  DIFFERENTIAL   CALCULUS  [Ch.  V. 

By  dividing  hy  x  —  a  and  then  letting  x  =  a,  it  follows 

that 

lim  f(x2  _f'(iay 
^  =  «</)(a;)      <i>'(a) 

The  functions  f'(a)-,  <^'(«)  will  in  general  both  be  finite. 
If  fi(a)  =  0,  </)'(a)  =^  0,  then  ^^  =  0. 

If  /'(a)  ^  0,  </,'(«)  =  0,  then  ^T^y  =  «• 

If  f\a)  and  <\)'(cl)  are  both  zero,  the  limiting  value  of 

f(x\ 

•iA_Z  is  to  be  obtained  by  carrying  Taylor's  development 

4>(x) 

one  term  further,  removing  the  common  factor  (x  —  a)^  and 

f"(a) 
then  letting  x  =  a.     The  result  is  '^ ,,     ^ • 

Similarly,  if  /(a),  /'(a),  /"(a)  ;  </,(a),  <^'(a),  </,"(«)  all 
vanish,  it  is  proved  in  the  same  manner  that 

«  =  «<^(2;)      <^"'(a)' 

and  so  on,  until  a  result  is  obtained  that  is  not  indeterminate 
in  form. 

Hence  the  rule: 

To  evaluate  an  expression  of  the  form  -,  differentiate  nume- 
rator and  denominator  separately ;  substitute  the  critical  value 
of  X  in  their  derivatives,  and  equate  the  quotient  of  the  deriva- 
tives to  the  indeterminate  form. 

Ex.  1.  Evaluate   ^—^^  when  6  =  0. 

Put  f(6)  =  \-cos6,      <i>(e)=e^; 

then  /'($)  =  sin  6,  <f>'(d)=2e, 

and  /'(0)=0.  <i>'(p)=0. 


72.]  INDETERMINATE  FORMS  123 

Again,  /"(0)  =  cos  6,  <{>"  (6)  =  2, 

/"(0)  =  1,  </."(0)=2, 

hence  ""^   l-«°«^     1 


Ex.  2.   Find 


e 

=  0 

^^ 

2 

n  e*  +  e" 

-'  +  2 

cosx- 

-4 

:0 

X* 

lim 

e'-e 

-*  — 2 

sinr 

x  =  0 

4x8 

lim 

e'+e 

-«_2 

cosx 

x  =  0 

12x2 

lim  , 

e^  — e" 

-  +  2 

sin  X 

x={) 

24  X 

_  lim 

e*+  e 

-x+O 

!  cosx 

X  =  0  -24 

=  -,  in  whichever  wav  x  =  0. 


Ex.  3.   Find     lim  ^-sinx  cosx 

Ex.  4.   Find     l'.*",  x«-2xS-4x2+9x-4. 
^  =  1       x*-2x»+2x-l 

In   this  example,  show  that  x  —  1  is  a  factor  of  both  numerator  and 
denominator. 

Ei.  5.   Find    lim  3tanx-3x-x«. 
x=  0  a* 

In  applying  this  process  to  particular  problems,  the  work 
can  often  be  shortened  by  evaluating  a  non-vanishing  factor 
in  either  numerator  or  denominator  before  performing  the 
differentiation. 

Ex.  6.    Find    lim  (^-4)Hana; 

_  lira  (x  —  iy  sec^  a:  +  2ra;— 4)tan  x 
""x  =  0  ^ 

=  16. 


124  DIFFERENTIAL   CALCULUS  [Ch.  V. 

The  example  shows  that  it  is  unnecessary  to  differentiate 
the  factor  (x  —  4)^  as  the  coefficient  of  its  derivative 
vanishes. 

In  general,  \if(x)  =  y^(x)')(^(x)^  and  if  '\^(a)  =  0,  '^(a^=^0. 
0(a)  =  O,  then 

lira  /(£)^^^^x^tt^ 

=  yC<^)     ,        1  since  'Jr(a)  =  0. 


Otherwise  thus : 


lim  -^(x^xi^^  ^   lira      .  .      lira  f(x)  ^  yj/Ca) 

Ex.7.   Fi»d^^„^27— -^,. 

Ex.8.   Find    lim(:»:-3)Mog(2-z)^ 
a;  =  l         8111  (a; -1) 

There  are  other  indeterminate  forms  than  - ;  they  are  — , 

0  00 

Qo  —GO,  0",  1*,  go".  The  form  0  —  0  is  not  indeterminate, 
the  value  of  the  function  being  evidently  zero. 

The  form  oo  —  oo  may  be  finite,  zero,  or  infinite. 

For  instance,  consider  Va;^  -f-  ax  —  x  for  the  value  x  =  cc  \ 
it  is  of  the  form  oo  —  go  ,  but  by  multiplying  and  dividing  by 

Va;2  +  ax  +  a;  it  becomes ,  which  has  the  form 

^yp-  -{-ax  4-  X 

—  when  x  =  cc). 

^         .  .   .   . 

Again,  by  dividing  both  terms  by  x,  it  takes  the  form 

and  this  becomes  ^z  when  a;  =  oo. 


V 


1+^4-1 


72-73.]  INDETERMINATE  FORMS  125 

73.  Evaluation  of  the  indeterminate  form  ^-. 
Let  the  function  "L- —  become  —  when  a;  =  a :  it  is  required 
tofind  j"^„-^. 

This  function  can  be  written 


f(x)^cf>Cxl 
<f>ix}      _1_ 

0 
which  takes  the  form  -  when  x  =  a,  and  can  therefore  be 

evaluated  by  the  preceding  rule. 

When  x  =  a, 

1  _'^\x) 


f(x') 
Dividing  through  by  "^^  :^,  it  becomes 

fCx)'^^ 

^4>(,x)'f(xy 

therefore  M     =^.  (2) 

This  is  exactly  the  same  result  as  was  obtained  for  the 
form  -  ;  hence  the  procedure  for  evaluating  the  indetermi- 
nate forms  X'  — '  is  the  same  in  both  cases. 

When  the  true  value  of  ,\  is  0  or  oo,  equation  CI)  is 
satisfied,  independent  of  the  value  of  ti_i^;  but  (2)  still 


126  DIFFERENTIAL   CALCULUS  [Ch.  V 

gives  the  correct  form  ;  for  suppose  ^  ^^^tS-J.  =  0  ;  and  con- 
sider the  function  ^ 
/(^          _f(x)  +  c^(x) 

which  has  the  form  —  when  x  =  a,  and  has  the  determinate 

00 

value  c,  which  is  not  zero  ;  hence  by  (2) 

lim  fix-)  +  c<f>(x')  ^fja)  +  c<^^(a)  ^  fja)  . 

therefore,  by  subtracting  <?, 

iim/M  =  /^ 

If  lim  /M  =  00,  then  J^i''^^  =  0,  which  can  be  treated 
as  the  previous  case. 

74.  Evaluation  of  the  form  oo  •  0. 

Let   the   function   be  ^(x)  •  -^(x),   such  that   ^(«)  =  oo, 

f(^a')  =  0. 

'\irCx')  0 

This  may  be  written    ^^  ^,   which    takes    the    form    - 

when  a  is  substituted  for  x,  and  therefore  comes  under  the 
above  rule.     (Art.  72.) 

75.  Evaluation  of  the  form  oo  —  oo.  There  is  here  no 
general  rule  of  procedure  as  in  the  previous  cases,  but  by 
means  of  transformations  and  proper  grouping  of  terms  it 

is  often  possible  to  bring  it  into  one  of  the  forms  -,    ■ — 

Frequently  a  function  which  becomes  oo  —  oo  for  a  critical 
value  of  X  can  be  put  in  the  form 

u      t 
V      w' 


73-75.]  INDETERMINATE  FORMS  127 

in  which  y,  w  become  zero;  and  this  equals 

uw  —  vt 
vw 

which  is  then  of  the  form  -. 

Ex.  1.   Find       .  t  (sec  a;  —  tan  x). 

This  expression  assumes  the  form  oo  —  oo,  but  can  be  written 
1     _  sin  X  _  1  —  sin  x 

cos  X        COS  X  COS  x 

which  is  of  the  form  -,  and  gives,  when  evaluated, 
_^  „  (sec  z  —  tan  x)  =0. 

lim 

Ex.   2.   Prove      ^  „  (sec"a;  —  tan"j;)  =  »,  1,  0  according  as 
n>2,     =2,     <2. 

EXERCISES 

Evaluate  the  following  expressions,  both  by  expansion  and  also  by 
differentiation;  examine  both  modes  of  approaching  limits: 

when        X  =  \. 
ar  =  0. 


^=2* 


x  =  \. 
3*-  15x2 +  24x-  10 

(Evaluate  also  without  the  use  of  derivatives.) 
«*-«-'-  2  X 


1. 

logx 
x-1 

;t 

e^-e-» 

smx 

3 

log  (2x2-1) 

tan  (x  —  1) 

4. 

log  sin  X 
(^-2x)^ 

5. 

x<-2x3  +  2x-l 

X  —  smx 

x-2 
f^X  -  1)»  -  1 


x  =  0. 
z  =  2. 


128  DIFFERENTIAL   CALCULUS  [Ch.  V 

• 1 

when        a;  =  0. 

x  =  0. 

x  =  0, 

X  =  i.. 


8. 

X  —  sin-'a; 

sin^a; 

9. 

a' -IF 

X 

10. 

tan  X  —  X 

X  —  sin  x 
^^     1  -  X  +  log  X 

1  ~  V2  X  —  x"^ 


_  2      m  sm  X  —  sm  mx  a;  =  0 

X  (cos  X  —  cos  mx) 

13. X  =  0. 

1  —  cos  mx 


_  .         yJa^  —  x^+  n  —  X 
14.    ___——— — I 

a/«^  -  ~  +  Vax  -  x2 


^  +  log(^) 


CSC  (7na~*) 
23.   ^"g(^+^> 


X  =  a. 


15.  xt-l+(x-l)t  J^l_ 

(x2  -  l)t  -X  +  1 

,_    x^cot^x  +  sinx  rt 

16.    X  =  u. 

X 

17.  (•e'-e-^)'=-2x2(-e»  +  g-')  j.  ^  q. 


18.  1  -W^x ^^0^ 

Vi  +  X  -  vTTx^ 

-n.    V"2  —  sin  X  —  cosx  ^      ir 

19.  ■ *  ~  7' 

log  sin  2  X  4 


20.                  V     e  /                                             X  =  0. 

tan  X  —  X 

2-1      secx  X  =  — 

sec  3  X  2 


22. 


76.]  INLETEHMlIiATE  FORMS  129 


INDETERMIJ^ATE  FORMS 

24.  e'sin-                                           when 

X 

Z  =  CO. 

25     **"  ^ 

tan  5  X 

-1- 

26.  ^-^^l^tanH 


29. 


32. 

--cot± 
z           2 

33. 

X              1 

X  —  1      log  X 

34. 

.  ^         cotsz 
siirx 

35. 

TT                       TT 

4  z     2  z  (ef '  +  1) 
36.  Prove  that  if /(a)  =  1,     <^(a)  =  1, 


lim    log/(z)  _  /'(a) 


37.  2«sin  — 
2- 


, —        (  TT     /a  — z) 

38.   Va^-^^cot\-^^yJ^ 

[z  +  sinz  —  4 sill- I 
{  3  +  cos  z  —  4  cos  ^  I 


X  =  a. 


27.  e   i  (1  -  log  z)  z  =  0. 

28.  log  (z  —  a)  tan  (z  —  a)  z  =  a. 

sec"z 


^  =  2- 


30.  (1  -  z)  tan  H  a:  =  1 

31.  ^(^^+^') 2^_  ^  =  0. 


Z  =  0. 
X=l. 

z  =  0. 
z  =  0. 


39.  4 li-  x  =  0. 


130  DIFFERENTIAL   CALCULUS  [Ch.  V. 

76.  Evaluation  of  the  form  1*. 

Let  the  function  u  =  [i/>(a^)]'''^"^^  assume  the  form  1"  when 
X  =  a. 

To  make  the  exponent  a  multiplier,  take  the  logarithm  of 
both  sides ;  then 

log  W  =  1/r  (a:)  .  log  (^  (a:)  =  i^^^. 


This  expression  assumes  the  form  -  when  x  =  a,  and  can 

be  evaluated  by  the  method  of  Art.  72. 

If  the  reduced  value  of  this  fraction  be  denoted  by  wi, 
then  log  u  =  m  and  u  =  e"*. 

Note.    The  form  1®  is  not  indeterminate,  but  is  equal  to  1. 

For,  let  [^  (2;)]*^^^  assume  the  form  1®  when  x  =  a. 

Put  w=  [</)(a:)]*(^), 

then  log  u  =  yfr  (x)  log  [<j>  (a;)], 

which  equals  zero  when  x  =  a; 
hence  log  u  =  0,        u  =  ^  =  1. 

77.  Evaluation  of  the  forms  0^,  oo*^. 
Let  [^(f)  (x)^'!'^^^  become  00®  when  x  =  a. 
Put  -     M=  [</>(aj)]*(^>, 

then  log  u  =y{r  (x)  log  <f)  (x)  =  — '^  !|        • 


This  is  of  the  form  ^,  and  can  be  evaluated  by  the  method 

GO 

of  Art.  72.     Similarly  for  the  form  O*.  . 

Note.     The  form  0"  is  not  indeterminate. 
For,  let  u  =  [<^(a;)]'''(^)  become  0±°°  when  x  =  a, 
then  log  u  =  ^^(x)  log  <^  (x)  =  T  00,  and  u  =  e^'^  =  0  or  oo. 
This  completes  the  list  of  ordinary  indeterminate  forms. 


76-77.]  INDETERMINATE  FORMS  131 

The  evaluation  of  all  of   them   depends  upon  the  same 
principle,  namely,  that  each  form  (or  its  logarithm)  may  be 

brought  to  the  form  -,  and  then  evaluated  by  differentiating 

numerator  and  denominator  separately.  In  finally  letting 
X  =  a,  the  two  directions  of  approach  should  be  compared, 
so  as  to  reveal  any  discontinuity  in  the  function. 


EXERCISES 

Evaluate  the  following  indetenninate  forms: 

1.  (cos  x)'^^'  when        x  =  0. 

2.  (cos  ax)"x^P*  x  =  0. 


'•  (!)""■ 


x  =  Q. 


4.  (1  -  a:)*  X  =  0. 

1 

5.  x^-'  X  =  1. 


M 


X  =  00. 


7.  (1  -  xy 

1         1 

In 

8 


1 
9.    (x  —  a)'~"  when  x  =  a  from  either  side. 


©IFF.  CALC.  — 10 


CHAPTER   VI 
MODE  OF  VARIATION  OF  FUNCTIONS  OF  ONE  VARIABLE 

78.  In  this  chapter  methods  of  exhibiting  the  march  or 
mode  of  variation  of  functions,  as  the  variable  takes  all 
values  in  succession  from  —  oo  to  +qo,  will  be  discussed. 
Simple  examples  have  been  given  in  Art.  19  of  the  use 
that  can  be  made  of  the  derivative  function  ^'(a;)  for  this 
purpose. 

The  fundamental  principle  employed  is  that  when  x  in- 
creases through  the  value  a,  ^{x)  increases  through  the 
value  <^(a)  if  0'(«)  is  positive,  and  that  <^(a;)  decreases 
through  the  value  <^(«)  if  ^'(ct)  is  negative.  Thus  the 
question  of  finding  whether  ^{x)  increases  or  decreases 
through  an  assigned  value  0(a),  is  reduced  to  determining 
the  sign  of  ^'(a). 

Ex.  1.   Find  whether  the  function 

increases  or  decreases  through  the  values  <^(3)  =  2,  ^(0)  =  5,  <^(2)  =  1, 
<^(—  1)  =  10,  and  state  at  what  value  of  x  the  function  ceases  to  increase 
and  begins  to  decrease,  or  conversely. 

79.  Turning  values  of  a  function.  It  follows  that  the 
values  of  x,  at  which  0(a:)  ceases  to  increase  and  begins  to 
decrease  are  those  at  which  ^'(a;)  changes  sign  from  positive 
to  negative  ;  and  that  the  values  of  a;,  at  which  ^{x)  ceases 
to  decrease  and  begins  to  increase,  are  those  at  which  ^' {x) 
changes  its  sign  from  negative  to  positive.  In  the  former 
case,  <^(a:)  is  said  to  pass  through  a  maximum^  in  the  latter, 
a  minimum  value. 


Ch.  VI.  78-79.]  VARIATIO^^   OF  FUNCTIONS 


133 


Ex.  2.   Find  the  turning  values  of  the  function 
<^  (x)  =  2  x3  -  3  x2  -  12  a;  +  4, 
and  exhibit  the  general  inarch  of  the  function  by  sketch- 
ing the  curve  y  =  4>(x). 

Here  <^'(x)  =  6  x-  -  6  z  -  12,  =  6(x  +  1)  (x  -  2), 
hence  <f>'(x)  is  negative  when  x  lies  between  -1  and  +2, 
and  positive  for  all  other  values  of  x.  Thus  <^(x)  increases 
from  X  =  -cotox=  -I,  decreases  from  x  =  -ltox  =  2 
and  increases  from  x  =  2  to  x  =  co.  Hence  <^(-  1)  is  a 
maximum  Ma.\ue  of  <f>(x),  and  <^(2)  a  minimum. 

The  general  form  of  the  curve  y  =  ^(x)  (Fig.  15)  may 
be  inferred  from  the  last  statement,  and  from  the  following  simultaneous 
values  of  x  and  t/ : 

^  =  -  =o,  -  2,  -  1,  0,      1,        2,      3,    4,  00. 


Fig.  15. 


y=  -00, 


16,  -  5,  36,  00. 

Exhibit    the    march    of    the 


0,     11,  4,  -  9,  - 

Ex.  3. 

function 

,^(x)  =  (x-l)l  +  2, 
especially  its  turning  values. 

2 1 

3 


Since     <^'(x) 


(X  -  l)i' 

hence  <^'(x)  changes  sign  at  x  =  1 ;  being 
negative  when  x<  1,  infinite  when  x  =  1, 
and  positive  when  X  >  1.  Thus^(l)=2 
is  a  minimum  turning  value  of  </>(x);  and 

the  graph  of  the  function  is  as  shown  in  Fig.  16,  with  a  vertical  tangent 

at  the  point  (1,  2). 


Fio.  16. 


Ex. 


Here 


4.   Examine  for  maxima  and  minima  the  function 
<^(x)  =  (x- 1)^  +  1. 


^  (X-  l)t 


hence  <t>\jc)  never  changes  sign,  but  is 
always  positive.  Thus  there  is  no  turning 
value.     The  curve  y  =  <^(x)  has  a  vertical 

tangent  at  the  point  (1,  1),  since-^  =  6'(x) 

dx 
is  infinite  when  x  =  1.    (Fig.  17.) 


Fig.  17. 


134  DIFFERENTIAL   CALCULUS  [Ch.  VI. 

80.  Critical  values  of  the  vrariable.  It  has  been  shown  that 
the  necessary  and  sufficient  condition  for  a  turning  value  of 
(f>(x)  is  that  ^'(x)  shall  change  its  sign.  Now  a  function  can 
only  change  its  sign  either  when  it  passes  througli  zero,  as  in 
Ex.  2,  or  when  its  reciprocal  passes  through  zero,  as  in 
Exs.  3,  4.  In  the  latter  case  it  is  usual  to  say  that  the 
function  passes  through  infinity.  It  is  not  true,  conversely, 
that  a  function  always  changes  its  sign  in  passing  through 
zero  or  infinity,  e.g.,  y  =  a^. 

Nevertheless  all  the  values  of  x,  at  Avhich  0'(a;)  passes 
through  zero  or  infinity,  are  called  critical  v-alues  of  x,  be- 
cause they  are  to  be  further  examined  to  determine  whether 
^'(x)  actually  changes  sign  as  x  passes  through  these  values  ; 
and  whether,  in  consequence,  4>(j^)  passes  through  a  turning 
value. 

For  instance,  in  Ex.  2,  the  derivative  (f>'(x)  vanishes 
when  re  =  —  1,  and  when  x  =  2,  and  it  does  not  become  in- 
finite for  any  finite  value  of  x.  Thus  the  critical  values  are 
—  1,  2  ;  and  it  is  found  that  both  give  turning  values  to 
<f)(^x').  Again,  in  Exs.  3,  4,  the  critical  value  is  a:=  1,  since 
it  makes  <f>'(x)  infinite,  and  it  gives  a  turning  value  to  ^(a;) 
in  Ex.  3,  but  not  in  Ex.  4. 

81.  Method  of  determining  whether  <f>'(x}  changes  its  sign 
in  passing  through  zero  or  infinity. 

Let  a  be  a  critical  value  of  x,  in  other  words  let  <f)'(a)  be 
either  zero  or  infinite,  and  let  A  be  a  very  small  positive 
number  ;  then  a  —  h  and  a  +  h  are  two  numbers  very  close 
to  a,  and  on  opposite  sides  of  it ;  thus  in  order  to  determine 
whether  (f)'(x')  changes  sign  as  x  increases  through  the  value 
a,  it  is  only  necessary  to  compare  the  signs  of  <f>'(a  +  h)  and 
^'(a  —  A).     If  it  is  possible  to  take  h  so  small  that  ^'(a  —h) 


80-81.] 


VARIATION  OF  FUNCTIONS 


135 


is  positive  and  ^'(a  +  h)  negative,  then  (f>'(x)  changes  sign 
as  X  passes  through  the  value  a,  and  0(a;)  passes  through  a 
maximum  value  ^(ji)-  Similarly,  if  ^'(«  —  h)  is  negative 
and  0'(a  +  Ji)  positive,  then  ^(x)  passes  through  a  minimum 
value  (\>{a). 

If  ^'(a  —  A)  and  ^'(a  +  A)  have  the  same  sign,  however 
small  h  may  be,  then  <^(a)  is  not  a  turning  value  of  </>(a;). 

Ex.  5.    Find  the  turning  values  of  the  function 

Here         <^'(^)  =  2  (-f  -  1)(^  +  1)^  +  3  (x  -  \)\x  +  1)2 
=  (x-l)(x+  l)2(5x-l), 

hence  *^\x)  passes  through  zero  at  x  =  —1,1,  and  1 ;  and  it  does  not 
become  infinite  for  any  finite  vahie  of  x. 

Thus,  the  critical  values  are  —  1,  ^,  1. 

When  X  —  —  \  —  h,  the  three  factors  of  <^'(-'^)  *'^^®  ^^^  signs  —  +  — , 
and  when  x  =  —  I  +  A,  they  become  —  +  — ; 

thus  <l>'(^)  does  not  change  sign  as  x  increases  through  —  1 ;  hence 
<^  (  —  1)  =  0  is  not  a  turning  value  of  <}>  (x). 

When  X  =  ^  —  k,  the  three  factors  of  </>'(•*■)  ^'"^  —  +  — , 

and  when  x  =  l  +  h,  they  become  —  +  +; 

thus  <l>'(x)  changes  sign  from  +  to  —  as  x  increases  through  ^,  and 
<f}  (I)  =  1  •  1  •••  is  a  maximum  value  of  </>(j')- 

Finally,  when  x=l  —  h,  the  three  factors  of  <f>'(x)  have  the  signs  — t-  +, 
and  when  x  =  1  +  A  they  become  +  +  + ; 

thus  <l>'(x)  changes  sign  from  —  to  +  as  x  increases  through  1,  and 
^(1)  =  0  is  a  minimum  value  of  «^(ar). 

Tlie  deportment  of  the  function  and  its  first  derivative  in  the  vicinity 
of  the  critical  values  may  be  tabulated  thus : 

+ 
inc. 

The  general  march  of  the  function  may  be  exhibited  graphically  by 
tracing  the  curve  y  =  <^(x)  (Fig.  18),  using  the  foregoing  result  and  also 
the  following  simultaneous  values  of  x  and  y  ; 

X  =  -  00,  -  2,  -  1,  0,      i,  1,     2,  oo. 

jy  =  _oo,  -9,      0,1,1^15,0,27,00 


X 

-1-h 

-1 

-1+h 

l-h 

i 

i+* 

1-h 

1 

4.'ix) 

+ 

0 

+ 

+ 

0 

- 

- 

0 

^(x) 

inc. 

infl. 
0 

inc. 

inc. 

max. 
1.1 

dec. 

dec. 

min. 
0 

136 


DIFFERENTIAL   CALCULUS 


[Ch.  VI. 


Fig.  18. 

Ex.  6.   Show  the  march  of  the  function 

^  (x)  =  sin^  x  •  CO.S  ar. 

^'(x)  =  2sina:cos2x  —  sin*  a? 

=  sin  X  (2  cos^  x  —  sin^  x), 

hence  the  critical  values  of  x  are  found  from  the  equations 

sin  X  =  0,  and  2  cos^  x  —  sin^  x  =  0,  or  tan  x  =  ±  V2. 

Thus  the  critical  values  of  x  are  x  =  0,  x  —  tt,  x  =  27r'"  and  x  =  ±a, 
IT  ±  a,  27r  ±  a,  ...  where  a  =  tan~^  y/2  =  .85  ...  radians. 

When  x  =  —  h,  the  factors  of  <f>'(x)  are  — ,  +, 
x  =  0,  0,  +, 

a:  =  + A,  +,  +; 

thus    <^'(x)  changes  from   —  to  +  as  x  increases  through   zero,  and 
<^  (0)  =  0  is  a  minimum  value  of  ff>  (x). 

When  X  =  TT  —  h,  the  factors  of  <f>'(x)  are  +,  +, 

X  =  TT,  0,   + , 

X  =  TT  +  A,  — ,  +  ; 

thus  <f>'(x)  changes  from  +  to  —  at  x  =  tt,  and  <f>  (tt)  =  0  is  a  maximum 
value  of  </>  (x). 

Similarly,  <f>'{x)  changes  from  —  to  +  at  x  =  27r,  and  <f)  (2  7r)  =0  is  a 
minimum  value  of  ^(ar). 


81-82.] 


VARIATION  OF  FUNCTIONS 


137 


Again,  when  x  =  a  —  h,  the  factors  of  «^'(x)  are  +,  +, 
x  =  a,  +,  0, 

X  =  a  +  h,  +,  -• 

(Observe  that  when  x  increases  to  a  +  h,  cos  x  diminishes,  and  sin  x 
increases;  thus  the  zero  factor  at  a;  =  a  becomes  negative  at  x  =a  +  h. 
Similarly,  it  becomes  positive  a,t  x  =  a  —  h.) 

Thus  <f>'(x)  changes  from  +  to  -  at  a;  =  a,  and  <^(x)  has  a  maximum 
value  at  ^(a). 

When  X  =  Tr  —  a  —  h,  the  factors  of  <f>'(^)  are  +,  — , 


X  =  IT  —  a, 

X  =  TT  —  a  +  h, 


+,0, 


i(Observe  that  since  tt  —  a  is  in  the  second  quarter,  diminishing  tt  —  a 
increases  the  sine  and  diminishes  the  cosine  numerically,  and  thus 
changes  the  zero  factor  to  negative.) 

Thus  <f>\x}  changes  from  —  to  +  as  a:  increases  through  tt  —  a,  and 
<f>(Tr  —  a)  is  a  minimum  value  of  <l>(x). 

It  may  be  shown  in  the  same  manner  that  ^(;r  +  a)  is  a  minimum, 
<^(2  TT  —  a)  a  maximum,  and  so  on. 

Combining  the  two  sets  of  results,  the  form  of  the  curve  is  found  to  be 
that  of  the  accompanying  figure  (Fig.  19). 


Fig.  19. 


82.  Second  method  of  determining  whether  (^'(x)  changes 
sign  in  passing  through  zero.  The  following  method  may  be 
employed  when  the  function  and  its  derivatives  are  continu- 
ous in  the  vicinity  of  the  critical  value  x  =  a. 

Suppose,  when  x  increases  through  the  value  a,  that  <\>'(x) 
changes  sign  from  positive  through  zero  to  negative.  Its 
change  from  positive  to  zero  is  a  decrease,  and  so  is  the  change 


138  DIFFERENTIAL    CALCULUS  [Ch.  VI. 

from  zero  to  negative  ;  thus  <^'(x)  is  a  decreasing  function 
at  a;  =  a,  and  hence  its  derivative,  (f>"(x),  is  negative  at 
x=a. 

On  the  other  hand,  if  <f)'ix)  changes  sign  from  negative 
through  zero  to  positive,  it  is  an  increasing  function,  and 
<f>"(x)  is  positive  at  a;  =  a;  hence  : 

The  function  <p(x')  has  a  maximum  value  ^(«),  when^'(a)  =  0 
and  4)"(a)  is  neyative ;  ^(x^  has  a  minimum  value  ^(a), 
when  ^'(a)=  0  and  <^"(a^  is  positive. 

It  may  happen,  however,  that  <^"(«)  is  also  zero. 

In  this  case,  to  determine  whether  0(a;)  has  a  turning 
value,  it  is  necessary  to  proceed  to  the  higher  derivatives. 
If  ^(x)  is  a  maximum,  <t>"{x}  is  negative  just  before  vanish- 
ing, and  negative  just  after,  for  the  reason  given  above  ;  but 
the  change  from  negative  to  zero  is  an  increase,  and  the 
change  from  zero  to  negative  is  a  decrease  ;  thus  ^''(x) 
changes  from  increasing  to  decreasing  as  x  passes  through  a. 
Hence  (f>"'(x')  changes  sign  from  positive  through  zero  to 
negative,  and  it  follows,  as  before,  that  its  derivative,  ^""(a;), 
is  negative. 

Thus  ^(«)  is  a  maximum  value  of  (f>(x)  if  ^'(«)  =  0, 
0"(a)=O,  ^"'(a)=0,  4>^^(a)  negative.  Similarly,  <^(a)  is 
a  minimum  value  of  <^(x')  if  ^'(a)=  0,  <^"(a)=  0,  0'"(a)=  0, 
and  ^'^(«)  positive. 

If  it  happen  that  <^'^(a)  =  0,  it  is  necessary  to  proceed 
to  still  higher  derivatives  to  test  for  turning  values.  The 
result  may  then  be  generalized  thus  : 

The  function  ^  (x)  has  a  maximum  (or  minimum^  value  at 
x  =  a  if  one  or  more  of  the  derivatives  (f)'(a^,  ^"(a),  0'"(«) 
vanish  and  if  the  first  one  that  does  not  vanish  is  of  even  order, 
and  negative  {or  positive^. 


82-83.]  VARIATION   OF  FUNCTIONS  139 

Ex.  7.   Find  the  critical  values  of  Ex.  5  by  the  second  method. 

«^"(1)  =  16,  hence  <^(1)  is  a  miuinium  value  of  <t>{x), 

</»"(  — 1)=0,  hence  it  is  necessary  to  find  ^'"(  —  1), 

<^"'(a:)=-Hx  +  l)(5a:-l)  +  o(a;  +  l)2+2(x  +  l)(ox-l)  +  2(x-l)(5z-l) 

+  10(x-l)(x+l)+.5(x  +  l)2+10(x-l)(a;+l). 

«^"'(  — 1)=24,  hence  ^(  —  1)  is  neither  a  maximum  nor  a  minimum  value 
of  <f>{x). 

Again,  ^"l-j=5(-  —  1|(-  +  1J  is  negative,  hence  <t>  {-)  is  a 
maximum  value  of  <j>(x). 

Ex.  8.    Examine  similarly  the  critical  values  of  Ex.  6. 
In  this  case  the  second  derivative  reduces  to 

<f>"(x)  =cos  x(2  cos^x  — 7  sin^x), 

hence  <^"(0)  is  jjositive,  <A"(7r)  is  negative ;  thus  «^(0)  is  a  minimum  and 
4>(ir)  a  maxinmm  value  of  ^(x). 

.\gain,  <^"(a)=cosa(2  cos^a— 7  siu'^a), 

but  a  satisfies  the  equation  2cos2a  — sin'*a=0,  hence  <f>"(a')  is  negative 
and  <li(a)  is  a  minimum  value  of  <^(r). 

Also  «^"(7r— a)  =  —  cosa(2cos2a  — 7  sin^a)  is  positive,  and  <^(7r  — a)  a 
minimiim  value  of  ^(x).     Similarly  for  the  other  critical  values  of  Ex.  6. 

83.  Conditions  for  maxima  and  minima  derived  from  Tay- 
lor's theorem. 

In  this  article,  as  in  the  preceding,  the  function  and  its 
derivatives  are  supposed  to  be  continuous  in  the  vicinity  of 
x=a  ;  otherwise  the  method  of  Art.  81  must  be  used. 

Let  ^(a)  be  a  maximum  value  of  ^(x);  then  it  follows 
from  the  definition  that  <f>(^a)  is  greater  than  either  of  the 
neighboring  values,  <^(a4-A),  ^(a  —  K),  when  h  is  taken  small 
enough.  Hence  ^(a  +  A)  — ^(a)  and  ^(a  — A)— <^(a)  are 
both  negative. 

Similarly,  these  expressions  are  both  positive  if  <^(a)  is  a 
minimum  value  of  <^(x). 


140  ^  DIFFERENTIAL   CALCULUS  [Ch.  VI. 

Let  <f)(x+h),  <^(x—K)  be  expanded  in  powers  of  h  by 
Taylor's  theorem  ; 

then       <^(:r  +  /0  =  «^(2:)  +  «/>'(a;)A+^^A2  +  ^^^^%t^/i^ 

If  X  be  replaced  by  a,  and  (}>  (a)  transposed,  there  results 
^(a  +  A)-0(a)  =  f(a)A  +  ^A^+'^"'^:+^^^A3. 

Zl  o! 

The  increment  h  can  now  be  taken  so  small  that  7i<f)'{a) 
will  be  numerically  larger  than  the  sum  of  the  remaining 
terms  in  the  second  member  of  either  of  the  last  two  equa- 
tions. Thus  ^(a  +  ^)  — <^(a)  and  ^(a  — A)— <^(a)  cannot 
have  the  same  sign  unless  ^'(a)  be  zero,  hence  the  first  con- 
dition for  a  turning  value  is  ^'(a)  =  0. 

In  this  case 

2!  31 

and  Ti  can  be  taken  so  small  that  the  first  term  on  the  right 
is  numerically  larger  than  either  of  the  second  terms,  hence 
<\>(^a-\-K)—<^(ci)  and  ^(^a  —  K)—<^(a)  are  both  negative  when 
^"(a)  is  negative,  and  both  positive  when  ^"(a)  is  posi- 
tive. 

Thus  ^(a)  is  a  maximum  (or  minimum)  value  of  <^(x) 
when  4*' (a)  is  zero  and  (j>'\a)  is  negative  (or  positive). 


83-84.]                       VARIATION   OF  FUNCTIONS  141 

In  case  it  should  happen  that  <^"{(i)  is  also  zero,  then 
4>(a  +  h}-(f>{a)  =  ^g^^   'h^  +  ^   ^^^ ^hS 

and  by  the  same  reasoning  as  before,  it  follows  that  for  a 
maximum  (or  minimum),  there  are  the  further  conditions 
that  <^"'(a)  equals  zero,  and  that  <^'^Ca)  is  negative  (or 
positive). 

Proceeding  in  this  way,  the  general  conclusion  stated  in 
the  last  Article  is  evident. 

Ex.  1.  Which  of  the  preceding  examples  can  be  solved  by  the  general 
rule  here  referred  to  ? 

Ex.  2.  Why  was  the  restriction  imposed  upon  <^'(a;)  that  it  should 
change  sign  by  passing  through  zero,  rather  than  by  passing  through 
infinity  ? 

84.  Application  to  rational  polynomials.  When  ^  (x)  is  a 
rational  polynomial,  its  derivative  <^\x)  is  of  similar  form. 
Let  the  real  roots  of' the  equation  ^'(a;)  =  0  be  a,  J,  e,  •••  Z, 
arranged  in  descending  order  of  algebraic  magnitude  ;  sup- 
pose, first,  that  no  two  of  them  are  equal ;  then  ^'(a;)  has 
the  form 

4>'(x)  =  (x-d)(x-  6)  (x  -  c) ...  (x  -  1}P,  (1) 

in  which  P  is  the  product  of  the  imaginary  factors  of  the 
polynomial  <^'(aj).  This  product  will  have  the  same  sign 
for  all  values  of  x,  and  by  giving  the  coefficient  of  the 
highest  power  of  x  in  <f>'Cx}  a  positive  value,  P  will  always 
be  positive,  by  the  theory  of  equations. 

Differentiating  (1)  with  regard  to  x,  and  putting  a;  =  a,  it 
follows  that 

<^"(a)  =  (a  -  6)  (a  -  c)  ..•  (a  -l)P, 


142  DIFFERENTIAL   CALCULUS  [Ch.  VI. 

but  a  —  5,  a  —  c,  •••  are  all  positive,  hence  <^"(a)  is  positive, 
aiid  therefore  <^(a)  is  a  minimum  value  of  <f>(x}. 

Again,      (f>"{b)  =  (b  -a){b-  c)  -(h-  l)P, 
but  6  —  a  is  negative,  and  the  remaining  factors  are  positive; 
hence  <f>"(J>}  is  negative,  and  <^(6)  is  a  maximum  value  of 
<f>(ix). 

Also  <f>"(c)  =  (e-a}{c-by"-(c-l)F, 

in  which  the  only  negative  factors  are  c  —  a,  e  —  b  ;  hence 
</)"(c)  is  positive  and  </>(<?)  is  a  minimum  value  of  <^(x)- 

Similarly,  the  fourth  root  (in  descending  order)  gives  a 
maximum,  and  so  do  the  sixth,  eighth,  •••,  while  tlie  first, 
third,  fifth,  •••  correspond  to  minima. 

Thus,  if  tlie  equation  <^'(x)  =  0  has  2w  real  roots,  all  of 
which  are  distinct,  the  function  <^(x)  has  n  maxima  and  n 
minima  occurring  alternately;  if  <^'(a;)  =  0  has2n  +  l  dis- 
tinct real  roots,  then  4*(x)  has  n-\-\  minima  and  n  maxima, 
the  latter  being  situated,  respectively,  between  successive 
minima. 

Next,  suppose  that  two  of  the  roots  are  each  equal  to  a ; 

then     <^'  (x)  =  (x  —  of'  i/r  (a:), 

and     <^"  (x)  =  (x-  ay  f  (x}  +  2  {x  -  a)  yjr  (a:), 

,fi"'  (x-)  =  (x-  ay  yjr"  (^x)  + -^  (x  -  a}  yjr' (x)  +  2  f  (x)  ; 
hence     </>'  (a)  =  0,     </>"  (a)  =.  0,     (f>"'  (a)  =  '2yfr  (a)  ; 

therefore  <f>  (a)  is  neither  a  maximum  nor  a  minimum. 

If  three  of  the  roots  of  (f)'  (a;)  are  each  equal  to  a,  it  is 
proved  similarly-  that  <j)  (a)  is  a  maximum  or  minimum  ac- 
cording as  yfr  (a)  is  negative  or  positive. 

These  conclusions  may  be  extended  to  the  cases  of  n  equal 
roots,  in  which  n  is  even  or  odd,  respectively. 

An  illustrative  example  was  given  in  Art.  81. 


84-86.]  VARIATION   OF  FUNCTIONS  143 

85-  The  maxima  and  minima  of  any  continuous  function 
occur  alternately.  It.  has  been  seen  that  the  maximum  and 
minimum  values  of  a  rational  polynomial  occur  alternately 
when  the  variable  is  continually  increased  or  diminished. 

Tliis  principle  is  also  true  in  the  case  of-  every  continuous 
function  of  a  single  variable  ;  for,  let  <f>  (a ),  cf)  (6)  be  two 
maximum  values  of  <^  (re),  in  which  a  is  supposed  less  than 
b ;  then  when  a;  =  a  +  A,  the  function  is  decreasing  ;  when 
x-=  b  —  h,  the  function  is  increasing,  h  being  taken  suffi- 
ciently small,  and  positive.  But  in  passing  from  a  decreas- 
ing to  an  increasing  state,  a  continuous  function  must,  at 
some  intermediate  value  of  x\  change  from  decreasing  to 
increasing,  that  is,  must  pass  through  a  mininuun.  Hence, 
between  two  maxima  there  must  be  at  least  one  minimum. 

It  can  be  similarly  proved  that  between  two  minima  there 
must  be  at  least  one  maximum. 

86.   Simplifications  that  do  not  alter  critical  values.     The 

work  of  finding  the  critical  values  of  the  variable,  in  the 
case  of  any  given  function,  may  often  be  simplified  by  means 
of  the  following  self-evident  principles. 

1.  Any  value  of  x  that  gives  a  turning  value  to  c0  (x) 
gives  also  a  turning  value  to  <f>  (x),  and  conversely,  when  c 
is  independent  of  x.  These  two  turning  values  are  of  the 
same  or  opposite  kind  according  as  c  is  positive  or  negative. 

2.  Any  value  of  x  that  gives  a  turning  value  to  <?  -|-  ^  (a:) 
gives  also  a  turning  value  of  the  same  kind  to  ^  (a:),  and 
conversely,  provided  c  is  independent  of  x. 

3.  Any  value  of  x  that  gives  a  turning  value  to  [^  (2;)]" 
gives  also  a  turning  value  to  <f>  (x),  and  conversely,  when  n 
is  independent  of  x.  Whether  these  turning  values  are  of 
the  same  or  opposite  kind  depends,  on  the  sign  of  w,  and  also 
on  the  sign  of  [«6  (x)Y. 


144  DIFFERENTIAL   CALCULUS  [Ch.  VL 

EXERCISES 

Find  the  critical  values  of  x  in  the  following  examples,  and  determine 
the  nature  of  the  function  at  each,  and  obtain  the  graph  of  the  function. 

1.    u  =  a;3  + 18x2+ 105x.  2.   u  =^  (x  -  \y{x  -  2yK 

3.  u  =  x{x-  l)«(x  +  1)3. 

4.  u  =  Ax"^  +  Bx  -\-  C ;  show  that  u  cannot  have  both  a  maximum 
and  a  minimum  value,  for  any  values  oi  A,  B,  C. 

5.  M  =  3  x^  —  2  X  +  4.    Show  that  a  cubic  function  has  in  general  both 
a  maximum  and  a  minimum  value. 

6.  w  =  2  X  +  4  —  X*.    Compare  the  graph  of  this  function  with  that 
of  exercise  5. 

7.  u=x'.  9.  M  =  ("  -  ^y. 

a-2x 
8.   u  —  — S —  10.   u  =  sin  2  X  —  X. 

X 

11.  Show  that  the  function  6  +  c  (x  —  o)^  has  neither  a  maximum 
nor  a  minimum. 

12.  w  =  sin^xcos'x.  14.   «  =  x  +  tanx. 

13.  w  =  sinx  +  cos2x.  15.   u  =  — \- e~^ . 

X 

87.  Geometric   problems   in  maxima   and  minima.      The 

theory  of  the  turning  values  of  a  function  has  important 
applications  in  solving  problems  concerning  geometric 
maxima  or  minima,  i.e.,  the  determination  of  the  largest  or 
the  smallest  value  a  magnitude  may  hava  while  satisfying 
certain  stated  geometric  conditions. 

The  first  step  is  to  express  the  magnitude  in  question 
algebraically.  If  the  resulting  expression  contains  more 
than  one  variable,  the  stated  conditions  will  furnish  enougli 
relations  between  these  variables,  so  that  all  the  others  may 
be  expressed  in  terms  of  one.  The  expression  to  be  maxi- 
mized or  minimized  can  then  be  made  a  function  of  a  single 
variable,  and  can  be  treated  by  the  preceding  rules. 


86-87.]  VARIATION   OF  FUNCTIONS  145 

Ex.  1.  Find  the  largest  rectangle  whose  perimeter  is  100.  Let  x,  y 
denote  the  dimensions  of  any  of  the  rectangles  whose  perimeter  is  100. 
The  magnitude  to  be  maximized  is  the  area 

u  =  xy,  (1) 

in  which  the  variables  ar,  y  are  subject  to  the  stated  condition 

2x  +  2y  =  100, 
I.e.,  y  =  50  -  x,  (2) 

hence  the  function  to  be  maximized,  expressed  in  terms  of  the  single 
variable  x,  is 

u  =  4>{x)  =  x (pO  -  x)  =  50x  -  x\  (3) 

The  critical  value  of  x  is  found  from  the  equation 

<^'(i)  =  50-2x  =  0, 

and  is  a;  =  25.  When  x  increases  tlirough  this  value,  <^'(i)  changes  sign 
from  positive  to  negative,  and  hence  </>(j')  is  a  maximum  when  x  =  25. 
Equation  (2)  shows  that  the  corresponding  value  of  y  is  25.  Thus  the 
maximum  rectangle  whose  perimeter  is  100,  is  the  square  whose  side  is 
25. 

Ex.  2.  The  sum  of  the  three  dimensions  of  a  rectangular  box  is  10, 
the  total  surface  is  34 ;  find  its  dimensions  so  that  its  volume  may  be  a 
maximum. 

Here  the  function 

u  =  xyz  (1) 

is  to  be  maximized,  the  three  variables  being  subject  to  the  two  condi- 
tions 

x  +  y  +  z  =  10,  (2) 

xy  +  xz  +  yz  =  17.  (3) 

Equation  (2)  multiplied  by  z,  subtracted  from  (3)  and  transposed, 

gives 

xy  =  17  -\0z  +  z\ 

by  means  of  which  the  variables  x  and  y  can  be  eliminated  from  (1), 

giving 

«  =  (17  -  10z  +  22)2. 

Hence  the  function  to  be  maximized  "by  varying  z  is 

,^(2)  =28  _  1022+  172, 

then  .^'(2)  =  3  2*  -  20  z  +  17  =  (z  -  1)  (3  2  -  17), 

f^"{z)  =6z-20; 


146 


DIFFERENTIAL    CALCULUS 


[Cn.  VI. 


hence  the  critical  value  z  =  \,  which  makes  ^'(z)  zero  and  <f>"(z)  negative, 
gives  to  ^(2)  the  maximum  value  8.  The  other  two  dimensions,  found 
from  (2)  and  (;i),  are  8  and  1.  The  second  ciitical  value,  z  =  5f,  makes 
</>"(2)  positive,  and  <^(2)  an  algebraic  minimum.  The  corresj)onding 
dimensions  are  5f,  —1^,  5|,  a  result  not  applicable  to  the  special  problem 
in  question.  Thus  the  required  dimensions  are  8,  1,  1.  Any  change  of 
these  dimensions  subject  to  the  given  conditions  will  lessen  the  volume. 

Ex.  3.  If,  from  a  square  piece  of  tin  whose  side  is  a,  a  square  be  cut 
out  at  each  corner,  find  the  side  of  the  latter  square  in  order  that  the 
remainder  "may  form  a  box  of  maximum  capacity,  with  open  top. 

Let  a;  be  a  side  of  each  square  cut  out,  then  the 
bottom  of  the  box  will  be  a  square  "whose  side 
is  a  —  2x,  and  the  depth  of  the  box  will  be  x, 
hence  the  volume  is 

v  =  x(a  —  '2  x)^, 

which  is  to  be  made  a  maximum  by  varying  x. 

dv 


Fig.  20. 


Here  — =  (a  —  '2  x)^  —  Ax(a—2x\ 

dx      ^  ^  ^  ' 

—  (a  — 2./:)(a  — 6a;). 


This  derivative  vanishes  when  a;=  ?;,  and  when  a;=  5.     It  will  be  found 

2.  0 

by  applying  the  usual  test,  that  x—  ^  gives  v  the  minimum  value  zero,  and 

a  "        2  a^ 

that  a;  =  «   gives  it  a  maximum  value  -^=-,  hence  the  side  of  the  square 

to  be  cut  out  is  one  sixth  the  side  of  the  given  square. 


Ex.  4.  Find  the  area 
of  the  greatest  rectangle 
that  can  be  inscribed  in 
a  given  ellipse. 

An  inscribed  rectangle 
"will  evidently  be  sym- 
metric with  regard  to 
the  principal  axes  of  the 
ellipse. 

Let  a,  h  denote  the 
lengths  of  the  semi-axes 
07l,05(Fig.21);  let2a:, 
2  2^  be  the  dimensions  of  an  inscribed  i-ectangle;  then  the  area  is 

u  =  4xy,  (1) 


Fig.  21. 


87.]  VARIATION  OF  FUNCTIONS  147 

in  which  the  variables  x,  y  may  be  regarded  as  the  coordinates  of  the 
vertex  P,  on  the  curve,  and  are  therefore  subject  to  the  equation  of  the 
ellipse 

It  is  geometrically  evident  that  there  is  some  position  of  P  for  whicli 
the  inscribed  rectangle  is  a  maximum ;  for  let  P  be  supposed  to  take  in 
succession  all  positions  between  A  and  B;  then  just  as  P  moves  away 
from  A  the  rectangle  begins  by  increasing  from  zero,  and  when  P  comes 
to  B  the  rectangle  ends  by  decreasing  back  to  zero ;  hence  there  must  be 
a  change  from  increasing  to  decreasing,  i.e.,  a  maximum,  for  at  least  one 
intermediate  position . 

The  elimination  of  y  from  (1),  by  means  of  (2),  gives  the  function  of 
X  to  be  maximized, 

46 


u  =ZJi X  Va^  -  x^.  (3) 

a 

By  Art.  86,  the  critical  values  of  x  are  not  altered  if  this  function  be 

divided  by  the  constant  — ,  and  then  squared.     Hence,  the  values  of  x 

a 
which  render  u  a  maximum,  give  also  a  maximum  value  to  the  function 

<f>  (x)  =  x2(a2  -  x2)  =  a2x2  _  X*. 

Here  <^' (x)  =  2  a2x  -  4  x^  =  2  z(a2  -  2  x^), 

<^"(x)  =2«--12x2; 

hence,  by  the  usual  tests,  the  critical  values  x  =  ±  -^  render  <f>  (x),  and 

V2 
therefore  the  area  «,  a  maximum.     The  corresponding  values  of  y  are 
given  by   (2),  and  the  vertex  P  may  be  at  any  of  the  four  points 
denoted  by 

V2  ^/2 

giving  in  each  case  the  same  maximum  inscribed  rectangle,  whose  dimen 
sions  are  aV2,  hV2,  and  whose  area  is  2ab,  or  half  that  of  the  circum- 
scribed rectangle. 

Ex.  5.  Find  the  cylinder  of  maximum  volume  that  can  be  cut  from 
a  given  prolate  spheroid. 

I^et  the  spheroid  and  inscribed  cylinder  be  generated  by  the  figure  of 
Ex.  4  revolving  about  OA  ;  then  the  volume  of  the  cylinder  is 

v  =  2irxy%  (1) 

DIFF.   CALC.  —  11 


148 


DIFFERENTIAL   CALCULUS 


[Ch.  VI. 


and  this  is  to  be  maximized  subject  to  the  condition 


-  +  ^=1; 


(2) 


hence 


V  =  — — X  (a''  —  x^), 
or 


and  by  Art.  86,  when  this  function  is  a  maximum,  so  is  the  function 
which,  according  to  the  usual  tests,  has  its  maximum  when  x  =  — ^. 


The  corresponding  value  of  y,  from  (2),  is 

maximum  volume  is 

4  Tralfl 


by/2 


hence,  from  (1),  the 


3V3' 


Q 


B 


or of  the  volume  of  the  prolate  spheroid. 

Ex.  6.  Find  the  greatest  cylinder  that  can  be  cut  from  a  given  right 
cone,  whose  height  is  h,  and  the  radius  of  whose  base  is  a. 

Let  the  cone  be  generated  by  the 
revolution  of  the  triangle  OAB 
(Fig.  22)  ;  and  the  inscribed  cylinder 
by  that  of  the  rectangle  A  P. 

Let  OA  =h,  AB  =  a,  and  let  the 
coordinates  of  P  be  (x,  y);  then  the 
function  to  be  maximized  is  iry'^(h—x^ 

subject  to  the  relation  ^  =  -. 
X      h 
Fig.  22. 

Ex.  7.  Find  the  area  of  the  greatest 

rectangle  that  can  be  inscribed  in  the  segment  of  the  parabola  y^  =  px, 

cut  off  Uy  the  line  x  =  a. 

Ex.  8.  What  is  the  altitude  of  the  maximum  cylinder  that  can  be 
inscri\)ed  in  a  given  segment  of  a  paraboloid  of  revolution  ? 

Ex.  9.  Find  the  greatest  right-angled  triangle  that  can  be  constructed 
on  a  given  line  as  hypothenuse. 

Ex.  10.  (iiven  the  vertical  angle  of  a  triangle,  and  its  area.  Find 
when  its  base  is  a  minimum. 

Ex.  11.  A  Norman  window  consists  of  a  rectangle  surmounted  by  a 
semicircle.  f>iven  the  perimeter;  required  the  height  and  breadth  of 
window  when  the  quantity  of  light  admitted  is  a  maxinmm. 


&7.]  VARIATION   OF  FUNCTIONS  149 

Ex.  12.  The  diameter  of  a  cylindrical  tree  is  a.  Find  the  strongest 
beam  that  may  be  cut  from  it,  assuming  that  the  strengfth  is  proportional 
to  the  breadth  multiplied  by  the  square  of  the  thickness. 

Ex.  13.  An  open  tank  is  to  be  constructed  with  a  square  base  and 
vertical  sides.  Show  that  the  area  of  the  entire  inner  surface  will  be 
least  if  the  depth  is  half  the  width. 

Ex.  14.  The  sum  of  the  perimeters  of  a  circle  and  a  square  is  fixed. 
Show  that  when  the  sum  of  the  areas  is  least,  the  side  of  the  square  is 
double  the  radius  of  the  circle. 

Ex.  15.  What  should  b^  the  ratio  between  the  diameter  of  the  base 
and  the  height  of  a  cylindrical  fruit  can  in  order  that  the  amount  of  tin 
used  in  constructifig  it  may  be  the  least  possible?  Solve  the  same 
problem  when  the  top  is  open. 

Ex.  16.  The  top  of  a  pedestal  which  sustains  a  statue  c  feet  in  height 
is  b  feet  above  the  level  of  a  man's  eyes.  Find  his  horizontal  distance 
from  the  pedestal  when  the  statue  subtends  the  greatest  angle. 

Ex.  17.  A  high  vertical  wall  is  to  be  braced  by  a  beam  which  must 
pass  over  a  parallel  wall  a  feet  high,  and  b  feet  distant  from  the  other. 
Find  tlie  length  of  the  shortest  beam  that  can  be  used  for  tlie  purpose. 

Ex.  18.  Determine  the  cone  of  minimum  volume  that  can  be  de- 
scribed about  a  given  sphere. 

Ex.  19.  Find  the  shortest  distance  from  the  point  (2,  1)  to  the 
parabola  y^  =  4:X. 

Ex.  20.  The  lower  corner  of  a  leaf,  whose  width  is  a,  is  folded  over 
so  as  just  to  reach  the  inner  edge  of  the  page;  find  the  width  of  the  part 
folded  over  when  the  length  of  the  crease  is  a  minimum. 

Ex.  21.  A  tangent  is  drawn  to  the  ellipse  whose  semi-axes  are  a  and 
b,  such  that  the  part  intercepted  by  the  axes  is  a  minimum ;  show  that 
its  length  is  a  +  6. 

Ex.  22.  A  person  being  in  a  boat  3  miles  from  the  nearest  point  on 
the  beach,  wishes  to  reach  in  the  shortest  time  a  place  5  miles  from  that 
point  along  the  shore;  supposing  he  can  walk  5  miles  an  hour,  l)ut  row 
only  at  the  rate  of  4  miles  an  hour,  find  the  place  where  he  must  land. 

Ex.  23.  A  slip  noose  in  a  rope  is  thrown 
around  a  large  square  post,  and  the  rope 
drawn  tiglit  in  the  direction  as  shown  in 
the  figure.  At  what  angle  does  the  rope 
leave  the  post  ?  Fig.  28. 


150       •  DIFFERENTIAL   CALCULUS  [Ch.  VI.  87. 

Ex.  24.  Show  that  just  before  and  after  a  turning  value  the  function 
passes  through  equal  values.  Apply  this  principle  to  give  geometrical 
solutions  to  Exs.  22,  23. 

Ex.  25.  Show  that  in  the  vicinity  of  a  turning  value  A/(x)  is  an 
infinitesimal  of  an  even  order  when  Ax  is  of  the  first  order.  When  is 
A/(a;)  of  the  third  order? 

Ex.  26.  A  rectangular  court  is  to  be  built  so  as  to  contain  a  given 
area,  and  a  wall  already  constructed  is  available  for  one  of  the  sides; 
find  its  dimensions  so  that  the  least  expense  may  be  incurred. 

Ex.  27.  The  work  of  driving  a  steamer  through  the  water  being  pro- 
portional to  the  cube  of  her  speed,  find  the  most  economical  rate  per 
hour  against  a  current  running  a  knots  per  houi*. 

W 

Ex.  28.    Assuming  that  the  current  in  a  voltaic  cell  is  C  = ,  E 

.  .  ^-^^ 

being  electromotive   force,  r  internal  resistance,  R  external  resistance, 

and  that  the  power  given  out  is  P  =  RC%  prove  that  P  is  a  maximum 

when  r  =  R.     Trace  the  curve  that  shows  the  variation  of  P,  as  R  varies. 

[Perry's  Calculus  for  Engineers.] 


CHAPTER   VII 
RATES  AND  DIFFERENTIALS 

88.  Rates.  Time  as  independent  variable.  Suppose  a  par- 
ticle P  is  moving  in  any  path,  straight  or  curved,  and  let 
«  be  the  number  of  space-units  passed  over  in  t  seconds;  then 
»  may  be  taken  as  the  dependent  variable,  and  t  as  the  in- 
dependent variable. 

Let  A«  be  the  number  of  space-units  described  in  the 
additional  time  A<  seconds ;  then  the  average  velocity  of  P 
during  the  time  A<  is  — •>  the  average  number  of  space-units 

described  per  second  during  the  interval. 

The  velocity  of  P  is  said  to  be  uniform  if  its  average 

A« 
velocitv,  — 1  is  the  same  for  all  intervals  A^     The  actual 

"    At 

velocity  of  P  at  any  instant  denoted  by  t  is  the  limit  which 
the  average  velocity,  for  the  interval  between  the  time  t  and 
the  time  t  +  At,  approaches  as  At  is  made  to  approach  zero 
as  a  limit. 

rri  lim    As      ds 

Ihus  v  =z=  .'    '    —  =  -- 

^^  =  ^A^       dt 

is  the  actual  velocity  of  P  at  the  time  denoted  by  t.  It  is 
evidently  the  number  of  space-units  that  would  be  passed 
over  in  the  next  second  if  the  velocity  remained  uniform 
from  the  time  t  to  the  time  t  +  1. 

It  may  be  observed  that  if,  for  the  word  "  velocity,"  the 
more  general  term,  "rate  of  change,"  be  used,  the  above 

161 


152  DIFFERENTIAL  CALCULUS  [Ch.  Vll. 

statements  will  apply  to  any  quantity  that  varies  with  the 
time,  whether  it  be  length,  volume,  strength  of  current,  etc. 
For  instance,  let  the  quantity  of  an  electric  current  be  G 

at  time  t,  and  C  +  AO  at  time  t  +  At;  then  the  average  rate 

'   .    AC 
of  change  of  current  in  the  interval  A^  is  — ?  the  averaore 

At  ^ 

increase  in  current  units  per  second  ;  and,  as  before,  the 
actual  rate  of  change  at  the  instant  denoted  by  t  is 

lim     AC^dC 
^'  =  0  At       dt' 

This  is  the  number  of  current-units  that  would  be  gained 
in  the  next  second  if  the  rate  of  gain  were  uniform  from  tlie 
time  t  to  the  time  t  +  1. 

Since  ^  =  ^  .  ^,  [Art.  21 

dx      dt     dt 

hence  -^  measures  the  ratio  of  the  rates  of  change  of  y 
(too 

and  of  X. 

It  follows  that  the  result  of  differentiating 

2/=/(^)  G) 

may  be  written  in  either  of  the  forms 

!=/'(-),  (2) 

The  latter  form  is  often  convenient,  and  may  also  be 
obtained  directly  from  (1)  by  differentiating  both  sides  with 
regard  to  t.  It  may  be  read :  the  rate  of  change  of  y  is 
f'(x)  times  the  rate  of  cliange  of  x. 

Returning  to  the  illustration  of  a  moving  point  P,  let  its 

coordinates  at  time  ^  be  a;  and  i/ ;  then  — -  measures  the  rate 

^  dt 


5.] 


BATES  AND  DIFFERENTIALS 


153 


of  change  of  the  a;-coordinate, 
and  may  be  called  the  velocity 
of  P  resolved  parallel  to  the 
a;-axis,  or  the  a:-component  of 
the  velocity. 

Similarly,  -^  is  the  y-compo- 

nent  of  velocity. 

These  three  rates  of  change  are  connected  by  the  equation 


V 

dy_ 

dt 

\         / 

^dt 

A^ 

^ 

_^daj 

P                       'dt 

X 

o 

^dt)       \dt)       \dt)' 


(4) 


Ex.  1.    If  a  point  describe  the  straight  line  3x  +  4y  =  5,  and  if  x 
increase  h  units  per  second,  find  the  rates  of  increase  of  y  and  of  *. 


Since 
hence 

and  when 


dy  _  3  dx 

dt~  -i  dt ' 

dt  dt         ' 

dt 


Ex.  2,  A  point  describes  the  parabola  y^  =  12  x,  in  such  a  way  that 
when  X  =  3,  the  abscissa  is  increasing  at  the  rate  of  2  feet  per  second  :  at 
what  rate  is  y  then  increasing?     Find  also  the  rate  of  increase  of  s. 


Since 


y2=12x, 

^^di-      dt' 

dy  _C)  dx  _ 
dt~y  dt 


hence,  when  z  =  3,  and 


dx  _  i)     dy^ 


dt 


dt 


6      dx, 
^Vixdt  ' 


±2. 


^^'^        (:ir=(l)'^(f)'   —ef  =  .v^  feet  pe.  second. 


154  DIFFERENTIAL    CALCULUS  [Ch.  VII. 

Ex.  3.  A  person  is  walking  towards  the  foot  of  a  tower  on  a  horizontal 
plane  at  the  rate  of  5  miles  per  hour ;  at  w  hat  rate  is  he  approaching  the 
top,  which  is  60  feet  high,  when  he  is  80  feet  from  the  bottom  ? 

Let  X  be  the  distance  from  foot  of  tower  at  time  t,  and  y  the  distance 
from  the  top  at  the  same  time  ;  then 

a;2  +  G02  =  y% 

dx  dy 

dt        ^  dt 

When  X  is  80  feet,  y  is  100  feet ;  hence  if  —  is  5  miles  per  hour,  -^ 
is  4  miles  per  hour. 

89.  Abbreviated  notation  for  rates.  When,  as  in  the  above 
examples,  a  time  derivative  is  a  factor  of  each  member  of  an 
equation,  it  is  usually  convenient  to  write,  instead  of  the 

symbols  — ,  — ,  the  abbreviations  dx  and  dy^  for  the  rates 
dt     dt 

of  change  of  the  variables  x  and  y.  Thus  the  result  of 
dilferentiating 

y=fix)  (1) 

may  be  written  in  either  of  the  forms 

g  =/'(:»),  (2) 

dy=f'(x)dx.  (4) 

It  is  to  be  observed  that  the  last  form  is  not  to  be  re- 
garded as  derived  from  equation  (2)  by  separation  of  the 

symbols  dy,  dx;   for  the  derivative  -^  has  been  defined  as 

dx 

the  result  of  performing  upon  y  an  indicated  operation  rep- 
resented by  the  symbol  — ;  and  thus  the  dy  and  dx  of  the 
symbol  -^  have  been  given  no  separate  meaning. 

The  dy  and  dx  of  equation  (4)  stand  for  the  rates  or  time 


88-89.]  RATES  AND  DIFFERENTIALS  155 

derivatives  -^  and  —  in  (8),  which  is  itself  obtained  from 
dt  dt        ^ 

(1)  by  differentiation  with  regard  to  i,  by  Art.  21, 

In  case  the  dependence  of  y  upon  x  be  not  indicated  by  a 
functional  operation/,  equations  (3),  (4)  take  the  form 

dy  _  dy  dx 
dt      dx  dt* 

dy  =  ^  dx. 
^      dx 

In  the   abbreviated  notation,   equation    (4)   of   the  last 

article  is  written 

d»^  =  dx^  +  dy^. 

Ex.  1.  A  point  that  is  describing  the  parabola  y^  =  2px  is  moving  at 
time  /  with  a  velocity  of  v  feet  per  second ;  find  the  rate  of  increase  of  the 
coordinates  x  and  y  at  the  same  instant. 

Differentiating  the  given  equation  with  regard  to  t, 

ydy  =  pdx, 

but  dx,  dy  also  satisfy  the  relation 

dx^  +  dy"^  =  v^ ; 
hence,  by  solving  these  simultaneous  equations, 

dx  =        ^        17,    dy  =        P        V,  in  feet  per  second. 

Ex.  2.  A  vertical  wheel  of  radius  10  ft.  is  making  50  revolutions  per 
second  about  a  fixed  axis.  Find  the  horizontal  and  vertical  velocities  of 
a  point  on  the  circumference  situated  30°  from  the  horizontal. 

Since  x  =  10  cos  0,    y  =  10  sin  6, 

rfx  =  -  10  sin  edO,  dy  =  10  cos  Odd, 
but  d6  =  100  TT  =  314.16  rad.  per  second, 

hence  dx  =  —  3141.6  sin  6  =  —  1570.8  feet  per  second, 

dy  =  3141.6  cos  6  =  2720.6  feet  per  second. 

Ex.  3.  Trace  the  changes  in  the  horizontal  and  vertical  velocity  in  a 
complete  revolution. 


156  DIFFERENTIAL   CALCULUS  [Ch.  VIL 

90.  Differentials  often  substituted  for  rates.  The  symbols 
dx,  dy  have  been  delined  above  as  tlie  rates  of  change  of  x 
and  y  per  second. 

They  may  sometimes,  however,  be  conveniently  allowed  to 
stand  for  any  two  numbers,  large  or  small,  that  are  propor- 
tional to  these  rates  ;  and  the  equations,  being  homogeneous 
in  them,  will  not  be  affected.  It  is  usual  in  such  cases  to 
speak  of  the  numbers  dx  and  dy  by  the  more  general  name 
of  differe7itial8,  and  they  may  then  be  either  the  rates  them- 
selves, or  any  two  numbers  in  the  same  ratio. 

This  will  be  especially  convenient  in  problems  in  which 
the  time  variable  is  not  explicitly  mentioned. 

Ex.  1.  When  X  increases  from  45°  to  4.5°  15',  find  the  increase  of 
logjpsin  X,  assuming  that  the  ratio  of  the  rates  of  cliange  of  the  function 
and  the  variable  remains  sensibly  constant  throughout  the  short  interval. 

Here  dy  =  .4343  cot  xdx  =  .4343  dx ; 

let  rfa:  =  15' =  .004363  radians; 

then  dy  =  .001895, 

which  is  the  approximate  increment  of  log^g  sin  x, 

but  logio  sill  45°  =  -  ^  log  2  =  -  .150515, 

logio  sin  45°  15'  =  -  .148612. 

Ex.  2.  Expanding  logj^sin  (x  +  li)  as  far  as  h^  by  Taylor's  theorem,  and 
then  putting  x  =  .785398,  h  =  .004363,  show  what  is  the  error  made  by 
neglecting  the  third  term,  as  was  done  in  Ex.  1. 

Ex.  3.   When  x  varies  from  60°  to  60°  10',  find  the  increase  in  sin  x. 

Ex.  4.  Show  that  logj^  x  increases  more  slowly  than  x,  when  x  >  logj„  e, 
that  is,  X  >  .4343. 

Ex.  5.  Two  sides,  a,  b,  of  a  triangle  are  measured,  and  also  the  in- 
cluded angle  C;  find  the  error  in  the  computed  length  of  the  third  side  c 
due  to  a  small  error  in  the  observed  angle  C. 

[Differentiate  the  equation  c^  =  a^  +  b^  -  2  ab  cos  C,  regarding  n,  b  as 
constant.] 


90.]  BATES  AND  DlF'FEliENriALS  157 

Ex.  6.  In  a  tangent  galvanometer  the  tangent  of  the  deflection  of  the 
needle  is  proportional  to  the  current.  Show  that  the  relative  error  in 
the  computed  value  of  the  current,  due  to  a  given  error  of  reading,  is 
least  when  the  angle  of  deflection  is  45°. 

Ex.  7.    The  en-or  in  the  area  A  of  an  ellipse,  due  to  small  errors  in  the 

semi-axes,  is  approximately  given  by =  — ■  -\ 

A         a         b 

Ex.  8.   The  side  of  an  equilateral  triangle  is  24  inches  long  and  is 

increasing  at  the  rate  of  two  inches  per  day ;  how  fast  is  the  area  of  the 

triangle  increasing? 

Ex.  9.  Find  the  rate  of  change  in  the  area  of  a  square  when  the  side 
b  is  increasing  at  a  ft.  per  second. 

Ex.  10.  In  the  function  y  =  2  x^  +  Q,  what  is  the  value  of  x  at  the 
point  where  y  increases  24  times  as  fast  as  x? 

Ex.  11.  A  circular  plate  of  metal  expands  by  heat  so  that  its  diameter 
increases  uniformly  at  the  rate  of  2  inches  per  second ;  at  what  rate  is 
the  surface  increasing  when  the  diameter  is  5  inches  ? 

Ex.  12.  What  is  the  value  of  x  at  the  point  at  which  x^  —  b  x-  +  VI  x 
and  x^  —  3  a:  change  at  the  same  rat«? 

Ex.  13.  Find  the  points  at  which  the  rate  of  change  of  the  ordinate 
y  =  a-8  —  6a:2  +  3x  +  5is  equal  to  the  rate  of  change  of  the  slope  of  the 
tangent  to  the  curve. 

Ex.  14.  The  relation  between"*,  the  space  through  which  a  body  falls, 
and  t,  the  time  of  falling,  is  s  =  IG^^j  show  that  the  velocity  is  equal 
to  32/. 

The  rate  of  change  of  velocity  is  called  acceleration;  show  that  the 
acceleration  of  the  falling  l)ody  is  a  constant. 

Ex.  15.  A  body  moves  according  to  the  law  s  —  cos  (nl  +  e) ;  show 
that  its  acceleration  is  negative  and  proportional  to  the  space  through 
which  it  has  moved. 


CHAPTER   VIII 

DIFFERENTIATION  OF  FUNCTIONS  OF  MORE  THAN  ONE 
VARIABLE 

In  the  previous  chapters  the  dependence  of  one  variable 
upon  another,  called  the  independent  variable,  has  been 
discussed.  The  mode  of  dependence  of  one  variable  upon 
two  others  will  next  be  considered;  and  the  relation  between 
the  dependent  variable  z  and  the  independent  variables  x 
and  y  will  be  expressed  in  tlie  form 

^=/(^,^).  (1) 

Examples  of  such  dependence  have  been  seen  in  coordi- 
nate geometry  of  three  dimensions ;  for  instance,  from  the 
equation  of  a  sphere  referred  to  its  center  as  origin 

x^  -\-  y^  -\-  z^  =  a\ 

any  one  of  the  variables  may  be  expressed  as  a  function  of 
the  other  two  ;  thus 

2  =  Va^  —  a^  —  y\ 

Conversely,  any  relation  of  the  form  (1)  can  be  exhibited 
graphically  by  taking  x,  y  as  coordinates  of  a  point  on  a 
horizontal  plane,  and  drawing  at  the  point  an  ordinate  to 
the  plane  to  represent  the  corresponding  value  of  the  func- 
tion z  ;  the  form  of  the  surface  of  which  (1)  is  the  equation 
will  represent  the  mode  of  variation  of  the  function. 

91.  Definition  of  continuity.  A  function /(.r,  y)  is  said  to 
be  continuous  in  the  vicinity  of  the  values  x  =  a,y  =  b;  when 

168 


Ch.  VIII.  91-92.]      FUNCTIONS    OF    TWO    VARIABLES  159 

/(a,  b')  is  -real,  finite,  and  determinate  (whetlier  unique  or 
multiple- valued)  ;  and  when  the  difference  /(a  +  h^h  -{•  k) 
—f(a,  6)  can  be  made  less  than  any  assigned  number  ^,  by 
taking  A,  k  small  enough,  independent  of  the  ratio  of  ^  to  A ; 
in  other  words,  when 

no  matter  in  what  way  h  iind  k  approach  their  limits. 

It  is  implied  that,  when  the  function  is  multiple-valued, 
attention  is  to  be  paid  to  the  correspondence  of  the  multiple 
values  in  the  two  members  of  this  limit-relation. 

In  geometrical  language  the  function /(a*,  y)  is  continuous 
at  x  =  a,  y  =  ^i  when  the  ordinate  of  the  surface  z  =  f(x^  y') 
drawn  at  the  point  (a -f- A,  h-\-k')  approaches  as  a  limit  the 
ordinate  drawn  at  the  point  (a,  V)  irrespective  of  the  direc- 
tion in  which  the  point  (a  ■\- h,  h  -\-  k')  moves  to  coincidence 
with  the  point  (a,  i).    [Cf.  Exs.  7,  9,  p.  182.] 

92.  Rate  of  variation.  Partial  derivatives.  The  most 
important  question  concerning  the  variation  of  a  continuous 
function  z  is :  what  is  the  rate  of  change  of  z  when  x  and  y 
vary  at  given  rates  ?  It  is  convenient  to  consider  first  the 
simpler  question  :  what  is  the  rate  of  change  of  z  when  x 
varies  at  a  given  rate,  and  y  remains  constant  ?  * 

In  this  case  a  is  a  function  of  the  single  variable  x^  and  its 

rate  of  change  is 

dz dz  dx  y-jN 

dt      dx  dt 

dz    ' 
in  which  it  is  to  be  understood  that  the  operation  —   is  per- 

formed  on  the  supposition  that  y  is  a  constant,  and  that  — 

is  the  rate  of  change  of  z  in  so  far  as  it  depends  on  the 
change  of  x.     To  indicate  tliese  facts  without  the  qualifying 


160  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

verbal    statements,    equation    (1)    will    be    written    in    the 
form 

'dt~~dx    dt'  ^""^ 

dz 
in  which  —  stands  for  the  a;-derivative  of  z  when  y  is  kept 
dx 

constant,  and  is  called  the  partial  derivative  of  z  with  regai-d 

djz 
to  x^  and  -^  denqtes  the  rate  of  change  of  z  in  so  far  as  it 
dt 

depends  on  the  change  of  x. 

Thus,  by  Art.  18,  the  partial  derivative  is  the  result  of  the 
indicated  operation 

dz^_    lim    A^_     lim    f(x  +  Aa;,  y')  —  f  (a:,  y') 

Similarly,  the  rate  of  cliange  of  z  when  x  is  kept  constant 
and  y  varies  at  a  given  rate  is  measured  by 

d^^dl,dy^  (^3) 

dt      by    dt 

d  z 
in  which    -^^   is  the   rate   of    change   of  z  in  so  far  as   it 

^^  dz 

depends  upon  the  change  of  y^  and  —   denotes  the  partial 

derivative  of  z  taken  with  regard  to  y^  that  is,  the  result  of 
the  operation  indicated  by 

a£  _    lim    A^  ^    lim    /  (x,  y  +  Ay)  -  /  (x,  y') 
dy       ^y  =  ^Ay       ^y  =  ^  Ay 

93.  Geometric  illustration.  Let  the  function  /(a-,  y~)  be 
represented  graphically  by  the  ordinate  of  a  surface  whose 
equation  is  z  =  f(x,  ?/)  and  let  a  vertical  section  be  taken 
parallel  to  the  plane  («,  a;)  at  a  given  distance  y  =  yi  from 
that  plane  ;  then  if  a  point  P  be  supposed  to  describe  on 
the  surface  the  contour  of  the  section,  the  ^-coordinate  will 
remain  constant,  and  the  value  of  the  varying  ordinate  z  will 


92-94.]  FUNCTIONS    OF    TWO    VARIABLES  161 

be  given  by  the  equation  z=f(x,y^.  If  the  rate  of  varia- 
tion of  X  at  any  instant  be  known,  the  corresponding  rate  of 
variation  of  z  is  given  by 

djZ  _dzdx_  df(x,  i/{)  dx 
dt       dx  dt  dx        dt 

which  may  be  called  the  rate  of  variation  of  the  ordinate 
in  the  a;-direction. 

The  partial  derivative  —  is  the  ratio  of  the  rates  of  in- 
dx 

crease  of  z  and  a;,  and  is  represented  geometrically  by  the 
slope  of  the  tangent  drawn  to  the  contour  at  P. 

Ex.  1.  A  point  P  on  the  surface  z  =  x^y  +  2  xy^  moves  in  the  plane 
y  =  2 ;  the  x-rate  is  10  feet  per  second ;  find  the  rate  of  change  of  z,  when 
P  is  passing  through  the  point  for  which  x  =  3,  and  also  the  direction 
and  velocity  of  the  motion  of  P. 

Differentiating  the  given  identity  with  regard  to  t,  y  being  kept  con- 
stant, —  =  (2xy  +  2y^)  —  =  20  —  =  200  feet  per  second,  and  the  slope 

of  the  tangent  at  P  in  the  plane  of  motion  is  20. 

The  velocity  of  P  in  the  curve  is  VlO''^+  200''^=  200.25  feet  per  second. 

Similarly,  if  P  move  on  the  surface  in  the  plane  x  =  a-j, 
the  rate  of  change  of  z  will  be  given  by 

dyZ ^dz  dy ^ df(x,  y)  dy 

dt      dy  dt  dy        dt 

dz 
and  — ,  the   ratio  of  the  rates  of  change  of  z  and  y,  will 

dy 
measure  the  slope  of  the  tangent  at  P  in  the  plane  of  motion. 

Ex.  2.  Find  for  the  same  surface  as  before,  at  the  point  for  which 
a:  =  3,  y  =  2  the  rate  of  change  of  z  in  the  y-directioii,  if  y  be  changing 
at  the  rate  of  5  feet  per  second,  x  being  kept  constant. 


dt.     ^  •^/  <lt 

in  the  direction  of  motion  is  33 


Here  '^  =  (x^  +  i  xy)  '^  =  33  ^  =  165  feet  per  second,  and  the  slope 
dt.  (It  dt 


94.   Simultaneous  variation  of  a?  and  y ;  total  rate  of  varia- 
tion of  z.      It  will  now  be  shown  that  when  x  and  y  vary 


162  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

simultaneously,  the  total  rate  of  change  of  z  is  the  sum  of 
its  separate  rates  of  change  as  x  and  1/  vary  alone ;  that  is, 

dt      dt       dt'  ^  ^ 

dz      dz  dx  ,   dz  dy  ^ox 

or  —  = 1 —'  (I ) 

dt      dx  dt      dy  dt 

For,  let  z  =f(x,  ^),  and  let  x,  y  start  at  the  values  a-^,  ^j, 
and  take  increments  Aa;,  A?/ ;  then  the  initial  value  of  z  is 
/(^i'  ^1)'  '^"^^  ^^^  f^n?iX  value  is  fix^  +  Aa:,  y^  -\-  A?/);  hence 
the  total  increment  of  z  is 

f(x^  +  Aa;,  yi  +  Ay}-f{x^,  y{). 

By  subtracting  and  adding  the  intermediate  value 

f(x^  +  Aa;,  2/1), 

in  which  x  alone  has  varied  from  its  original  value,  the 
total  increment  of  z  may  be  written  as  the  sum  of  two 
partial  increments  in  the  form 

Az  =  lf(x^  +  ^x,  y^  +  Ay)-f(x^  +  Ax,  ^j)] 

+  [/(a^i  +  Ax,  yO-fC^v  yi)]  5 

the  latter  being  the  increment  of  /(a;,  y')  as  a;  changes  from 
ajj  to  ajj  +  Aa;,  y  remaining  constant,  and  the  former  being 
the  further  increment  of  the  function  as  x  remains  at  the 
value  x^  +  Aa;  while  y  changes  from  y^  to  y^  +  Ay. 

The  result  of  dividing  by  A^,  the  increment  of  i,  may 
be  written 

Az^fQx^  +  Ax,  y^  +  Ay)  -fjx^  4-  Aa:,  y^  Ay 
At  Ay  At 

fjx^  +  Aa;,  y^)-i\x^,  y^)  Ax 
"^  Aa;  At 


94]  FUNCTIONS    OF    TWO    VARIABLES  163 

Taking  limits  as  At,  Ax,  Ay,  Az,  all  approach  the  limit 
zero,  and  remembering  that  by  Art.  92, 

•^^  ^   -^^ — -^  ^  ^  ^^^  =  — ,  taken  at  a;  =  a:,,  y  —  y.. 

Ax  dx  ^   ^      ^^' 

f(x,+Ax,  y,^Ay)-f{x,+Ax,  y,^  ^d^^  ^^^^^  ^^  ^  ^     ^^^ 
Ay  by  ^ 

=  — ,  taken  at  x  =  x^, 
dy 

it  follows  that,  at  any  values  of  x  and  y,  for  which  the  func- 
tion and  its  partial  derivatives  are  continuous, 

dz  _dz  dx      dz  dy 
dt      dx  dt       dy  dt 

In  the  abbreviated  rate  notation,  equations  (2),  (3)  of 
Art.  92,  and  (1),  (2)  of  \vt.  94,  are  respectively, 

d^  =  ^  dx,         dyZ  =  ~  dy, 
dx  dy 

dz  =  djZ  +  djz  —  —  dx  ^ dy. 

dx        ^y 

Ex.  1.  A  particle  moves  on  the  spherical  surface  x^  +  y^  +  2^  =  0*  in 
a  vertical  meridian  plane  inclined  at  an  angle  of  60°  to  the  plane  (2^). 

If  the  x-component  of  its  velocity  be  ^a  per  second,  when  x  —  \a,  find 
the  ^-component,  the  2-coniponent,  and  the  resultant  velocity. 


Since  2  =  Va^  —  x^  —  y^, 


y/a^  —  x^  —  y'^      y/a^  —  x'^  —  y^ 
but  since  dx  =  ^a,  and  the  equation  of  the  given  meridian  plane  is 

y  =  x  tan  60*^,  hence  dy  =  dxVS  =  —  Vri,  and  y  =  ^^.    Therefore 
9  >  :/  10  4 

dz  =  -  -^  -  ^  =  _  ^^  =  _  .115a  in  feet  per  second. 
2V3      2  15 


Also,   ds  —  Vrfx^  +  dy^  +  dz'^  —  - — -  =  0.23  a  in  feet  per  second. 

7.0 

DIFF.   OALC.  —  12 


164  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

95.  Language  of  differentials.  The  results  of  the  preced- 
ing articles  may  be  stated  thus  : 

The  partial  z-differential  due  to  the  change  of  x  is  equal 
to  the  2;-differential  multiplied  by  the  partial  2;-derivative. 

The  partial  z-differential  due  to  the  change  of  1/  is  equal 
to  the  ^-differential  multiplied  by  the  partial  ^/-derivative. 

The  total  z-differential  is  equal  to  the  sum  of  the  partial 
2-differentials. 

One  advantage  in  keeping  the  equation  in  the  differential 
form  is  that  it  may  be  divided  when  necessary  by  the  differ- 
ential of  any  other  variable  s,  to  which  x  and  «/  are  related, 
and  then,  remembering  that  the  ratio  of  two  differentials 
(or  rates)  may  be  expressed  as  a  derivative,  the  equation 
becomes 

dz  _dz  dx      dz  dj£ 
ds      dx  ds      By  ds 

Ex.  1.   Given  z  =  axy"^  +  bx'^y  +  cjfi  +  ey, 

dz  =  {ay"^  +  2  bxy  +  3  cx^^dx  +  (2  axy  +  bx^  +  e)dy. 

Ex.  2.   Given      2  =  x",      d^  =  yx»~Mx,       dyZ  —  x"  log  x  dy, 
dz  —  yx^'^dx  +  x»  log  x  dy. 

„            ^ .                                ,  V          .        X  dv  —  V  dx 
Ex.  3.   Given  «  =  tan-^  ^>       du  =  — % ^ — • 

X  x^  +  y^ 

Ex.  4.  Assuming  the  characteristic  equation  of  a  perfect  gas,  vp  =  Rt, 
in  which  v  is  volume,  p  pressure,  t  absolute  temperature,  and  R  a  con- 
stant ;  express  each  of  the  differentials  dv,  dp,  dt,  in  terms  of  the  other 
two. 

Ex.  5.  Being  given  that  in  the  case  of  air,  R  =  96,  when  p  is  measured 
in  pounds  per  square  foot,  v  in  cubic  feet,  and  t  is  centigrade ;  and  letting 
t  =  300,  p  =  2000,  V  =  14.4 ;  find  the  change  in  p  when  t  changes  to  301, 
and  V  to  14.5,  supposing  that  p  changes  uniformly  in  the  Interval. 
[Perry's  Calculus  for  Engineers,  p.  138.] 

Since  vdp  +  pdv  =  Rdt,     dv  =  .1,     dt  =  1; 

hence  dp  =  -  7.22. 


95-96.]  FUNCTIONS  OF  TWO   VARIABLES  165 

The  actual  increment  of  p  will  be  a  little  different  from  this,  and  is 
easily  found  by  direct  computation  to  be  —  7.17. 

The  difference  in  the  results  is  analogous  to  the  difference  between 
the  ordinate  of  a  surface  and  tlie  ordinate  of  its  tangent  plane,  taken 
near  the  common  ordinate  of  the  point  of  contact. 

96.  One  variable  a  function  of  the  other.  When  there  is 
a  deiinite  relation  connecting  tlie  variables  x  and  y,  the 
equation 

dz  =  —  dx  -\ — -  dv 
dx  ay 

may  be  divided  by  the  differential  of  either  variable, 

then  ^  =  ^  +  ^^.  (1) 

dx      dx       dy  dx 

It  is  here  well  to  note  the  difference  between  •—  and  — 

ox  dx 

The  former  is  the  partial  derivative  of  the  functional  ex- 
pression for  z  with  regard  to  a;,  on  the  supposition  that  y 
is  constant.  The  latter  is  the  total  derivative  of  z  with 
regard  to  a;,  when  account  is  taken  of  the  fact  that  y  varies 
with  X. 

It  is  to  be  observed  that  the  implied  assumption  in  Art.  94,  that  the 
variables  x  and  y  have  at  any  instant  some  definite  numerical  rate  of 
change,  is  only  equivalent  to  assuming  that  they  vary  in  some  continuous 
manner.  They  need  not  on  that  account  be  expressible  as  definite  func- 
tions of  the  time,  or  have  any  fixed  relation  of  dependence  upon  each 
other.  On  the  other  hand,  a  fixed  relation  of  dependence  is  not  pre- 
cluded, for  Art.  94  only  assumes  that  x,  y  take  the  increments  Ax,  Ay  in 
the  time  A<,  without  inquiring  whether  one  of  the  increments  may  not 
be  determined  by  the  other,  or  whether  they  may  not  both  arise  from 
the  increment  of  some  other  hidden  variable.  The  supposition  that  the 
letters  x,  y  are  independent  in  forming  the  partial  derivatives  is  only  a 
convenient  algebraic  rule  or  artifice  for  obtaining  the  coefficients  of  the 
differentials  dx,  dy\  and  does  not  imply  the  physical  independence  of 
the  magnitudes  denoted  by  these  letters.  Thus  the  "  independence  "  is 
formal,  or  operational,  rather  than  physical. 


166  DIFFERENTIAL   CALCULUS  [Ch.  Vlll. 

The  value  of  -^  is  to  be  obtained  by  differentiatiiifir  the 

functional  relation  between  x  and  «/.  Jf  this  relation  ex- 
presses y  as  an  explicit  function  of  x,  the  right  hand  mem- 
ber of  (1)  can  then  be  expressed  in  terms  of  x  ahjne  ;  and 
the  result  will  be  the  same  as  if  z  had  been  first  reduced  to 
the  form  of  a  function  of  the  single  variable  x,  and  then 
differentiated  with  regard  to  this  variable. 

Ex.  1.  A  point  moves  on  the  surface  z  =y(x,  ?/)  in  the  curve  deter- 
mined by  the  cylindrical  surface  y  =  <l>  (j)  ;  express  dz  in  terms  of  dx. 

Ex.  2.    If  ^  =  tan  i  -^,  and  i  x^  +  y' =  \,  find  — • 
2  X  dx 

Ex.3.  A  point  moves  on.tlie  curve  of  intersection  of  the  surfaces 
z  =  (fi(x,  y),  z  —f(x,  y)  ;  find  the  mutual  ratio  of  the  rates  dx  :  dy :  dz 
at  any  point  x,  y,  z. 

For  shortness,  denoting  partial  derivatives  by  subscripts, 
dz  —f^lx  +f^ly  =  <j>ydx  +  <l>/iy, 
hence  dx :  dy  :  dz  =  <f>2  — /2  :./'i  —  <^i  '■f\<l>j  —J'i^v 

97.  Differentiation  of  implicit  functions ;  relative  variation 
that  keeps  z  constant.  An  important  special  question  is 
how  to  vary  x  and  i/  so  as  to  keep /(a:,  y)  from  varying.  If 
z  =f  (x^  y)  =  constant, 

then  dz=:-^dx  +  -^dy  =  0, 

dx  By 

hence  the  relative  rates  of  change  of  x  and  y  are  given  by 
the  equation  df 

dy  _      dx 

dx         df 
dy 

Ex.  1.  If  X  pass  through  the  value  2  at  the  rate  of  5  units  per  second, 
at  what  rate  must  y  pass  through  the  value  3  in  order  to  keep  the  func- 
tion x^y  +  3x^2  constant? 

Since       d  (c^y  +  3  xy"^)  =  (2  xy  +  3  y^)  dx  +  (x^  +  6  xy)  dy  =  0, 
hence  39  dx  -\-  iO  dy  =  0,     dy—  —  4 J  units  per  second. 


96-97.]  FUNCTIONS   OF  rH'O    VARIABLES  167 

Ex.  2.  Defining  the  elasticity  of  a  gas  as  the  limit  of  the  ratio  of  an 
increment  of  pressure  to  the  corresponding  relative  decrement  of  volume, 
find  e,  the  elasticity  of  a  perfect  gas  under  constant  temperature. 

and  by  differentiating  vp  =  Rt,  keeping  t  constant, 

vdp  +  pdv  —  d,     -4-  =  —  —,     hence  e  =  p. 
do  V  ^ 

As  a  geometrical  illustration,  let  a  section  of  the  surface 
z=f(x,  y)  be  made  by  the  plane  z  —  c,  then  for  all  points 
on  the  contour  of  the  section 

2  =f(x,  y)  =  c, 

and  if  a  point  describe  the  contour,  the  a;-rate  and  the  ^-rate 

will  be  in  the  ratio  —  -r-  :  ^-;  and  this  ratio  will  measure 
oy     ax 

the  slope  of  the  tangent  to  the  contour  with  reference  to  the 
plane  zx. 

~2        ^2        -2  . 

Ex.   A  particle  is  moving  on  the  ellipsoid  — ^-i-^-  —  =1  at  the  point 
,  a*      b'^     c^ 

X  =-,  V  =-,  z  —  -^;  find  the  relative  rates  of  x  and  of  y  so  that  the 

2'  ^      2'         V2 
rate  along  z  may  be  zero. 

Since  ^Jl  +  yAl^Q,    hence  ^=-^  =  --^. 

Similarly,  if  a  point  whose  coordinates  are  x,  y  move  in  a 
plane  so  as  to  keep  the  function  f(x^  «/)  constant,  then  it 
describes  a  curve  whose  equation  is  f(x,  y^  =  c,  hence  the 
differentials  dx^  dy  are  connected  by 

^ldx  +  ^fdy  =  0,  (1) 

dx  ay 

and  the  slope  of  the  direction  of  motion  is  given  by 

^  =  _^  .  ^.  (2) 

dx  dx  '  dy 


168  DIFFERENTIAL   CALCULUS  ICu.  VIIl. 

In  all  such  cases  either  variable  is  an  implicit  function  of 
the  other,  and  thus  the  last  equation  furnishes  a  rule  for 
finding  the  derivative  of  an  implicit  function.  In  many 
examples   in   practice  it  is  preferable   to   equate  the  total 

differential  to  zero,  as  in  (1),  and  then  solve  for  -^. 

dx 


Ex.1. 

Given  x^  +  ij^  +  3  axy  =  c,    find  -^. 

dx 

Since 

(3  x2  +  3  ay)  dx+  {iy^  +  Z  ax)  dy  =  0, 

dy         x2  +  ay 
dx~     2/2-1-  ax 

Ex.2. 

f{ax  +  by)=c;     ^=af'(ax  +  by);     g= 

--bf(ax  +  by);    g  = 

a 
~b 

Ex.3. 

If  0x2  ^2hxy  +  hy"^  +  2gx  +  2fy  +  c  = 

:  0,  find  ^. 
dx 

Ex.4. 

Given  X*  -  ?/*  =  c,  find  -j-' 

98.  Functions  of  more  than  two  variables.  All  of  the 
methods  of  this  chapter  are  applicable  to  functions  of  three 
or  more  variables. 

Let  u  =f{x,  y,  z), 

then  it  can  be  shown,  as  in  Art.  94,  that 

du_dudx      du  dy      du  dz  .^  -. 

dt      dx  dt      dy  dt      dz  dt* 

or  in  the  abbreviated  notation, 

du  =  ^-:^dx  +  ^-^dy  +  ^-^dz.  (2) 

dx        dy         dz  ^  ^ 

No  simple  geometric  representation  can  be  given  of  a 
function  of  three  variables,  but  there  are  many  examples  in 
physics  of  functions  of  the  three  coordinates  of  a  point ;  for 
instance,  the  potential  u  produced  at  a  point  (a:,  y,  z)  by  a 
given  distribution  of  fixed  attracting  bodies,  is  a  definite 
function  of  the  variables  x,  y,  z,  and  equation  (2)  gives  the 
rate  of  change  of  the  potential  as  the  point  (x,  y,  z)  changes 
its  position  in  any  direction. 


97-99.]  FUNCTIONS   OF  THREE  VARIABLES  169 

First  let  it  be  required  to  vary  (x,i/,  z)  so  that  the  potential 
u  =f(x,  i/,z^  shall  remain  constant;  then  the  point  must 
remain  on  the  equipotential  surface  whose  equation  is 
f(x,  y,  z)  =  c,  and  the  differentials  of  a;,  y,  z  are  connected 
by  the  relation 

|rf,+.|rfy+|rf.  =  0.  (3) 

In  such  cases  z  is  an  implicit  function  of  the  two  variables 
r,  1/ ;  and  its  differential  is  expressed  in  terms  of  their 
differentials  by  the  last  equation. 

Ex.  1.  If  the  "  characteristic  equation  "  of  a  substance  be 
/{p,  V,  i)  =  constant 

prove         m  •  m  ■  m    =-1. 

\(it  Jecon.    \dv/pcon.  ^flp/,  eon. 

Since  Mdp+^dv  +  ^dt  =  0, 

dp  av  dt 

hence,  if  rfi;  =  0,   -f-  =  —  -^  :  -^-     Similarly  for  dp  =  0,  etc. 
dt  dt      dp  J  I 

Ex.  2.  In  the  equation  c^  =  a^  +  ft'^  —  2  aA  cos  C,  referred  to  in  exercise 
6  of  Art. 90,  find  the  error  in  c  arising  from  given  small  errors  in  a,h,  C , 
all  the  errors  being  supposed  so  small  that  their  squares  can  be  neglected ; 
within  the  required  degree  of  accuracy. 

99.   One  or  two  relations  between  the  three  variables  x,  y,  z. 

Again,  if  the  point,  instead  of  moving  on  the  surface  u  =  con, 
move  on  some  other  surface  defined  by 

z  =  4>(x,y^,  (1) 

then  dz  =  ^^dx  +  ^^dy,  (2) 

and  (2)  of  Art.  98  becomes,  by  elimination  of  dz, 

\dx       dz    dx)  \dy      dz   dyJ   ^  ^  ^ 


170  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

The  point  has  then  only  two  degrees  of  freedom,  indicated 
by  the  independent  differentials  dx,  dy.  If  the  point  be 
further  restricted  to  the  curve  determined  on  the  latter 
surface  by  the  cylinder 

y^y^Cx),  (4) 

then  dy  =  -^'(x)  dx, 

and  (3)  becomes  by  elimination  of  dy,  and  division  by  the 
single  independent  differential  dx, 

dx      dx       dz  dx      \dy       dz   dyj 

This  derivative  could  also  be  obtained  by  eliminating  z 
and  y  before  differentiation.  The  function  u  in  terms  of 
the  single  variable  x  is  then 

u=f(x,ylr(x^,  <j>{x,y}r(x')}'). 

The  latter  method  is  usually  longer,  and  is  not  applicable 
at  all  when  equations  (1)  and  (4)  are  replaced  by  implicit 
relations  that  cannot  be  solved  for  one  of  the  variables. 

Ex.  1.   u  =  ea  (y  —  2),  2  =  a  sin  ^,  y  =  a  log  -;  find  — 

X  a  dx 

Ex.  2.  If  u  =  f(x,y,z);  and  if  x,  y,  z  are  connected  by  the  two 
relations  <f>{x,  y,  2)  =  Cj,  i{/(x,  y,  z)  =  C2;  find  du  in  terms  of  dx. 

Differentiating,  and  denoting  partial  derivatives  by  subscripts  for 
shortness, 

du  =  f^dx  +  /g  dy  +  /g  dz, 

0  =  <^j  dx  +  (f>2dy  +  <^3  dz, 
0  =  j/^j  dx  +  1/^2  dy  +  ij/sdz; 
hence,  by  elimination  of  dy,  dz. 

du  (<^.^l/r.,  -  <f>.^lP^)  =  dx  [/,  (<^2l/^3  -  <f>^\l/o)  +  /2(<^,j</',  -  <^,</'..,)  +  f;>X<l>l^2  '  ^i'/'l)  ]  • 

Geometrically  speaking,  the  point  (^x,  y,  z)  moves  on  the 
curve  of  intersection  of  two  surfaces  and  has  therefore  only 
one  degree  of  freedom. 


99-100.]  FUNCTIONS  OF  THREE   VARIABLES  171 

Thus  the  variation  of  a  single  independent  coordinate 
is  sufficient  to  determine  the  variation  of  the  other  coordi- 
nates, and  of  the  function  u  itself. 

100.  Euler's  theorem.  Relation  between  a  homogeneous 
function  and  its  partial  derivatives.  Let  u  =f(x,  y^  z)  be 
a  homogeneous  function  of  x,  y,  2,  of  degree  w;  then 

du  ,      du  ,      du 

X Vy  -Z f-2 =  WW. 

bx  ay  dz 

For,  let  u=Ax^y^zy  +  Bx^^'y^'z^'  +  ••• 

where  a  +  /3  +  7  =  «'  +  /3'  +  7'  =  •••  =  w. 

—  =  aAx^-^y^zy  +  a'Bx^'-^y^'zy  +  •••, 
dx 


x-^  =  uAx^y^z^  +  a'Bx'^'y^'zy'-^-' 
ox 


Similarly, 

y^  =  ^Ax^y^zy  +  fi'Bx^'y^'zy'  +  — , 
dy 

0  -^  =  jAxy^zy  +  'y'Bx°^y^'zy'-\-"*. 
oz 

Adding  these  three  equations, 

du  ,      du  ,     du 

X- — h  y \-  z  — 

ox         dy         dz 

=  («  +  /S  +  r^')Axf'y^zy  +<«'  +  ;8'+  '^')B7f-'y^'zy'  +  ... 
=  n  {Ax^y^zy  +  Bx^-'y^'z"^  +  •••) 
=  nu. 

The  theorem  can  be  extended  to  functions  of  any  number 
of  variables. 

If  a  function^  Jiomogeneous  in  several  variables^  he  differ- 
entiated partially  with  reyard  to  each  of  them;  then  each 
•partial  derivative  he  multiplied  hy  the  variable  with  regard 


172  DIFFERENTIAL   CALCULUS  [Ch.  VIII.  100. 

to  which  the  derivative  was  taken;  and  all  these  products 
added;  the  result  is  n  times  the  original  function  ;  where  n 
is  the  degree  of  the  function. 

This  is  known  as  Euler's  theorem.* 

EXERCISES 
Verify  Euler's  theorem  for  the  following  expressions : 


1. 

a:4  +  3  2:V-7x/. 

4. 

x^  —  3  x^y  - 

-y" 

2. 

(xi  +  ?/i)(x»  +  2^»). 

5.   tan-i^. 

X 

3. 

sin-. 

6.   fix^  ,  y   sj 

r^y. 

*  y^     x^        X 

Prove  the  following  identities  : 

1.  u  =  log(e=^  +  cv),  |^  +  |^  =  1- 

ax     dy 

2.  u  =  log(xH^'  +  28-3xy2),        ^  +  ^  +  ^=: — § . 

dx     dy      52      x  +  y  +  z 

3.  u=xyy',  x^-Vy^^{x-\-y^-\ogu)u, 

Qx        dy 

4.  u  =  sm-'^(xyz),  —  —  —  =tan2usec u. 

3x  5^^  ^2 

5.  u  =  log(tanx  +  tan3^+tan2),    sin  2x^  +  sin2y  — +sin2  2  — =2. 

dx  dy  dz 

6.  M  =  e'sin3/  +  e>'sinx,  \^\  +i  —  \=e^+e-y-\-2e'^+'ism(x-\-ij). 

7.  «  =  log(x+V5H?),  x|H  +  ^^  =  l. 

dx        dy 

8.  U:.l0g^,  rf„=_^_lop^. 

9.  M  =  logy',  du  =  logy  dx+  -  dy. 

10.  u=3^«i°»,  rf«  =  i/"°'cosxlogyrfx+-^i2_^6f^. 

♦Leonard  Euler  (1707-1783),  one  of  the  most  eminent  mathematicians 
of  the  eighteenth  century. 


CHAPTER  IX 
SUCCESSIVE  PARTIAL  DIFFERENTIATION 

101.   Successive  differentiation  of  functions  of  two  variables. 

Let  z=f  (x^  y),  in  which  a;,  y  are  functions  of  another  vari- 
able ^,  which  may  conveniently  be  thought  of  as  time.  As 
the  rate  of  change  of  z  is  usually  variable,  it  is  sometimes 
useful  to  have  an  expression  for  the  rate  of  change  of  this 

rate.     Just  as  —  is  the  rate  of  change  of  2,  so  — ( — 1  is  the 
dt  ^  dt\dt) 

rate  of  change  of  — ,  and  it  is  written  -— r.      This  rate  of 
dt  dt^ 

the  rate  may  conveniently  be  called  the  acceleration. 

It  will  now  be  shown  that  the  expression  for  the  «-acceler- 
ation  involves  the  a;-acceleration,  the  y-acceleration  and  also 
the  squares  and  products  of  the  a;-rate  and  the  y-rate,  each 
with  a  certain  coefficient. 

It  was  proved  in  Art.  94  that  the  2-rate  is 

dz      dz  dx     dz  dy  .^. 

dt      dx  dt      By  dt  ^  ^ 

Differentiating  each  term  of  this  identity  with  regard  to  i, 

d^z  _  d  fdz  \dx      dz  d^x      d  fdz  \dy      dz   ^y  ^         ^2\ 
lx^~Jv^x)'dt      Vxdt^      dt\dy)dt      dy~di^''         ^^ 

but,  since  —  is  itself  a  function  of  two  variables,  hence,  by 
ox 

Art.  94, 

dt \dx)  ~  dx \dx) dt      dy  \dx) dt ' 
178 


174 


DIFFER  EN  TIA  L   CA  L  C  UL  US 


[Ch.  IX. 


also 


dt\di/J      dx\dy)  dt      By  \dy)  dt ' 


d(2      dx\dx)\dt) 


+ 


±(dz\     ±fdz\~ 
_dy\dxj      dx\dyj_ 
d_fdz_\fdy\^ 


..^d^^y 
dx   dfi       dy  dfi  ^   ^ 


hence  (2)  becomes,  by  substitution    and  slight  re-arrange- 
ment, 

dx  d/jj^ 

dt    dt 

dz    d^x 

The  successive  partial  x-  and  ^-derivatives 

l.(^      L(^      1,(^      l.f^, 
dx\dxj       dx\dy/      dy\dx/      dy\dyj 

which  appear  as  coefficients  of  the  squares  and  product  of 
the  rates  will  be  denoted  by  the  symbols 

d'^z         d^z  d^z         d^z 

dx^' 


dx  dy       dy  dx       dy^ 


dh 


In  other  words,  — .    will  stand  for  the  operation  of  differ- 
dx^ 

entiating  2  twice  in  succession  with  regard  to  x,  on  the  sup- 

d^z 

position   that    y   is   constant ;    and will    denote    the 

dxdy 

operation  of  differentiating  z  first  with  regard  to  y,  on  the 
supposition   that   x   is   constant,   and    then    differentiating 
the  result  with  regard  to  x  on  the  supposition  that  y  is  con- 
stant; and  similarly  for'the  other  expressions. 
With  this  notation,  eq.  (3)  is 


d^z^d^zfdx\2    r  dh 


+ 


d^z 


dt^      dx\dt)       \_dy  dx      dxdy_ 


dx  dy 
dt  dt 


'Sz  fdyS^     dzd^x     dz  d^y 
dy\dtj      dx  dt^      dy  dt^ ' 

The  coefficient  of  the  product  of  the  rates  will  be  further 
simplified  by  the  theorem  of  the  next  article. 


101-102]     SUCCESSIVE  PARTIAL  DIFFERENTIATION  175 

102.   Order  of  differentiation  indifferent. 
Theorem.     The  successive  partial  derivatives 

By  Bx  Bx  By 

are  equal  for  any  values  of  x  and  y  in  the  vicinity  of  which 
2  and  its  first  and  second  partial  x-  and  y-derivatives  are 
continuous. 

For,  let  z  =  f(x,y^\  and  first  change  x  into  x  +  h,  keeping 
y  constant,  then  by  the  theorem  of  mean  value,  the  incre- 
ment of  the  function  is  equal  to  the  increment  of  the  variable 
multiplied  by  the  derivative  taken  for  some  value  interme- 
diate between  x  and  x  -\-  h;  that  is, 

f(x-¥hy^-f(x,y)  =  h  ±f{x^eKy^  [0<^<1. 

Bx 

Now  let  y  change  to  y  +  k,  x  remaining  constant,  and  take 
the  increment  of   the   function  on   the   left;    then   by   the 

theorem  of  mean  value  applied  to  — f(x-\-dh,y^  as  a  func- 
tion of  y,  for  the  increment  k, 

lf(x  +  A,  y  +  k)-f(x.  y  +  Ar)]  -  U(x  +  h.  y)  -f(x,  y)] 

=  kh  —  —f(x  +  eh,y  +  0,k). 
By  Bx 

Next  let  these  increments  be  given  in  reversed  order ;  then 

[fix  +  h,y  +  k)-f(x  +  K  3/)]  -  lf(x,  y  +  k)-f(ix,  y}] 

=  hk^^f{x-\-e^h,y  +  ejcy, 

BxBy 
hence 

A I  Ax  ^OKy^  9,k)  =  A  ^f(x  +  0^h,  y  +  ^3^) 
By  Bx  Bx  By 

for  any  values  of  h  and  k  within  the  range  around  the  point 
(a;,  y)  within  which  all  the  functions  mentioned  are  con- 
tinuous. 


176  DIFFERENTIAL   CALCULUS  [Ch.  IX. 

When  A,  k  approach  zero, 

X  -{-  dh,  y  +  d-Jc^  and  x  +  6^h,  y  +  6Jc 
approach  (x,  y),  and,  by  Art.  91, 

f(x  +  eh,  y  +  e^k\  f(x  +  e^K  y  +  ^3^) 

approach /(a;,  y);  and  similarly  for  the  derivatives;  hence 
or,  since  f{x,  y)  =  2, 


By  dx      dx  dy 

Cor.  1.  It  follows  directly  that  under  corresponding 
conditions  the  order  of  differentiation  in  the  higher  partial 
derivatives  is  indifferent.     In  other  words,  if  u  and  all  its 

partial   derivatives  are   continuous,   the   operations  — »  -— 
^  ,.  '^  dx    dy 

are  commutative. 

Tjj  9%  d^u  S^u 


dx  dy  dx      dx^  By      dy  dx^ 

CoE.  2.     Equation  (3)  may  now  be  written  in  the  simple 
form 

'd^~d^\dtj  dxdydidt       dyAdt) 

du  d^x     du  d^y  fA^ 

dx  dfi      dy  dt^ 

or,  if  the  independent  variable  t  is  not  expressed, 


du  jq     ,   du 


<Pu  =  ^(^.)^+2^<«.<J,  +  |^«i^> 


+2Hrf2^+^d2„.  (5) 

dx  dy 


102.]  SUCCESSIVE  PARTIAL  DIFFERENTIATION  177 

If  X  be  taken  as  independent  variable,  then  t  is  to  be  re- 
placed by  X ;  and  since  -—  =  0,  the  equation  becomes 

dx^      dx^         dxdydx      dy^\dxj       dy  dx^ 

Similarly,  if  y  be  taken  as  independent  variable,  and  x  be 
a  function  of  y,  then 

tPit  _  dhi I'dx'^     o   drhi    dx      d^u      du^x  .rr^ 

dy^      dx^  \dyj         dx  dy  dy      dy^      dx  dy^ 


EXERCISES 

1.  Verify  that  -^^  =  -^^,  when  u  =  xhj^. 

dxdy     dyd^ 

2.  Verify  that  —S- = ,  when  u  =  x^y  +  xy*. 

dx  dy^     dy^ dx 

3.  Verify  that  ~ —  =  -^ — ,  when  u  =  y  log  (1  +  xy\. 

dx  dy     dy  dx 

4.  In  Ex.  3  are  there  any  exceptional  values  of  x,  y  for  which  the 
relation  is  not  true  ? 

5.  Given  «  =(x*  +  y^y,  verify  the  formula 

x^  — h"  xy  -^ h  y^  - —  =  0. 

dx^  dx  dy         dy^ 

6.  Given  u  =  (a;^  -\.  y^p,  show  that  the  expression  in  the  left  member 

of  the  equation  in  Ex.  5  is  equal  to  — • 

4 

7.  Given  u=(x^  +  y^  +  z^y^ ;  prove  that  ^+^^^  +  ^^  =  0. 

8.  Given  «  =  sec  (y  +  ax)  +  tan  (y  —  ax)  ;  prove  that  —  =  a^  ^■ 

9.  Given  «  =  sin  x  cos  y ;  verify  that  — - — -  = =  -^-^ — • 

dy'^  dx'''     dx  dy  dx  dy     dx'^  dy'^ 

10.    Given  «  =  (4  aft  -  c2)-^  ;  prove  that  ^  =   ^^  ■ 

dc^      da  dh 


178  DIFFERENTIAL   CALCULUS  [Ch.  IX. 

11.  If  u  =  sin  V,  I-  being  a  homogeneous  function  of  degree  n  in  x  and 

V,  determine  the  value  ol  x—  +  y  — • 
"  dx      ^  dy 

12.  If  M  =  tan-i —    ^y  show  that  -r^  =  (1+ a:^  +  y2)-|  and 

Vl  +  x2  +  2/2  3a:  dy 

that    ^*"  =      i^^.y 

aa:2a.y2      (1  +  3.2  +  ^2)! 


103.  Extension  of  Taylor's  theorem  to  expansion  of  func- 
tions of  two  variables. 

Taylor's  theorem,  as  developed  in  Chapter  IV,  relates 
only  to  functions  of  one  variable,  but  it  can  be  readily 
extended  to  functions  of  any  number  of  variables  in  a 
manner  first  shown  by  Lagrange. 

Let /(a:,  y)  be  a  function  of  the  two  variables  x,  y,  which, 
with  its  first  2  n  partial  derivatives,  as  to  x  and  y,  is  finite 
and  continuous  for  all  values  of  x,  y  within  a  certain  por- 
tion of  the  coordinate  plane.  It  is  required  to  expand 
f(x  +  A,  y  +  ^)  in  a  series  of  ascending  powers  of  h  and  of  k. 

Using  an  auxiliary  variable  t^  let 

x'  =  X  +  ht,        y'  =  y  -^  kt,  (1) 

then  /(a;',  y')  =f(x  +  ht,  y  +  kt)  =  ^(0,  say ;  (2) 

the  development  of  -^(0  in  powers  of  t  is,  by  Maclaurin's 
theorem, 

+  Z!(^^»,  [o<^<i.    (3) 

n  I 
whence,  putting  <  =  1, 

f(^x  +  h,y  +  k}  =  F(H)-)  +  F'CO}+-^^^  +  ^^.     (4) 


102-103.]      SUCCESSIVE  PARTIAL  DIFFERENTIATION         179 

To  express  ^(0),  ^'(0)  •••in  terms  of  A,  k,  first  find  ¥'((), 
F"(t),  F"'(t),  '"  by  successive  differentiation  of  (2);  then 

but,  from  (1),  —-  =  h,       -^  =  k,  hence 

^  ^  dt  dt 

likewise,  ~     ^^'         V 

^^        dt\dx'J        dAdy'J 

Ux'2  (^i       dx'dy'   dt)        \dx'dy'  dt       dy'^  dt)' 
then  putting  — —  =  A,       -f-  =  A;, 


dx^  bxby  ay' 

Similarly, 


^"(0=^^li^+2A^^,+^^- 


Now  when  t  is  replaced  by  zero  in  these  derivatives,  a/,  y' 
reduce  back  to  x,  y;  hence 

FaO')=f(x,y-), 

F\Q-)=hf+kf^ 
ax         by 

dar  &xay  ay^ 

oar  axray  axoy^  oy^ 

DIFF.  CALC.  — 13 


180  DIFFERENTIAL   CALCULUS  [Ch.  IX. 

and  when  t  is  replaced  by  0  in  F"(t),  x'  and  y'  become 
X  +  0h  and  y  +  6k^  hence 

^  ^     ZJ\rJ  dx"-'du'- 


in  which  (    ]  stands  for  the  binomial  coefficient 


rj  rl(ji  —  r}! 

Therefore  (4)  becomes 

21 V     dx^  dxdy^       dy'^r 

(n-V)\L^\    r    J  dx^-'-^df 

n\ZJ\r)  dx^-'dy'-  ^  ^ 

which  is  the  desired  form  of  expansion, 

104.  Significance  of  remainder.  This  expansion  is  useful 
only  for  those  values  of  re,  y,  h,  k,  for  which  the  last  term, 
called  the  remainder  after  the  (n  —  l)st  powers  of  h  and  k, 
can  be  made  as  small  as  desired  by  taking  n  large  enough. 

105.  Form  corresponding  to  Maclaurin's  theorem.  The 
expansion  of  a  fur.ction  of  two  variables  in  a  series  of  powers 
of  these  variables  can  be  readily  obtained  from  the  last 
equation. 

If  it  be  desired  to  expand  /  (a:,  «/)  in  the  vicinity  of  a,  6, 


103-105.]      SUCCESSIVE  PARTIAL  DIFFERENTIATION         181 

let  X  =  a,  1/  =  b  ;  h  =  x  —  a^k  =  i/  —  b,  then  equation  (^5)  of 
Art.  103  becomes 

nl^Vr         ^     ^       ^  da'^-'db'- 


36' 


in  which  -^  denotes  that  / (x.  y)  is  to  be  differentiated  with 
da  J  \  ^ifJ 

regard  to  x,  and  that  x  =  a^  y  =  b  are  then  to  be  substituted 

in  the  derivative  ;  and  so  for  the  other  symbols. 

In  particular,  if  a  =  0,  6  =  0,  the  expansion  of  /  (x,  y)  in 

powers  of  x^  y  becomes 


/(.„)=/(o,o)+.[g^+,[|]^ 

+ ...  +      1     y  (^  -  lV-.-y^'-/(0-  0) 

r=0  "^ 

These  theorems  for  expansion  can  be  readily  extended  to 
functions  of  any  number  of  variables. 


182  DIFFERENTIAL   CALCULUS  [Cii.  IX.  105. 

EXERCISES 

1.  Expand  e*  sin  (x  +  y)  iu  powers  of  x  and  y. 

2.  Expand  (x  +  y)*  in  powers  of  x  and  y. 


3.  Expand  s/x  +  h  tan  (y  +  k)  in  powers  of  h  and  ^-j  and  express  the 
form  of  the  remainder  after  two  terms,  when  h  =  1,  k  =  1. 

4.  Expand  the  function  x'^  +  y^  +  z^  —  4:X  +  Qy  —  2z  —  11  in  powers 
of  a;  —  2,  ^  +  3,  z  —  1. 

5.  Arrange  the  function 

3x^-5y^  +  4:x-7i/  +  ll 
in  powers  of  z  —  2,  y  +  3. 

6.  Transform  the  equation 

x^  +  y'  —  3x1/  =  1 
to  parallel  axes  with  the  point  (1,  2)  as  origin. 

7.  What  kind  of  discontinuity  has  the  function     ^    — ^  at  the  point 

x  +  2y 
(0,  0)  ?    Show  that  it  may  have  any  value  between  2  and  3,  depending 
on  the  ratio  oi  y  to  x  as  they  approach  zero.     Illustrate  geometrically. 

8.  Write  down  an  expression  for  the  error  in  the  approximate  equa- 
tion 

(cf.  Ex.  5,  p.  156 ;  Ex.  5,  p.  164 ;  Ex.  2,  p.  169). 

9.  When  x  =  a  and  y  =  a,  prove  that,  without  discontinuity, 

(x  -  y)a»  +  (y  -  a)x''  +  (a  -  a:)^"  ^      «(n  -  1)    ^_g 
i^-yXy  -<i)(^a-x)  2 

10.  If  «  =/(x,  y)  =  c,  prove,  by  (2),  p.  167,  and  (6),  p.  177,  that 

d^y  _  dx^ \dy)      '  dx  By  dx  dy     dy^  \dx) 
dx^  (dny 

\dyJ 


CHAPTER   X 
MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  TWO  VARIABLES 

106.  Definition  of  maximum  and  minimum  of  a  function 
of  two  variables.  A  continuous  function  z  =  (j)  (x,  y)  has 
a  maximum  value  ^  (a,  6)  for  a;  =  a,  y  =  J  if ,  as  the  variables 
pass  in  any  manner  through  the  values  a  and  6,  the  function 
hitherto  increasing,  ceases  to  increase  and  begins  to  de- 
crease ;  the  function  has  a  minimum  value  ^  (a,  H)  if  it 
ceases  to  decrease,  and  begins  to  increase,  for  every  varia- 
tion of  X  and  y  through  the  values  a  and  h. 

This  fact  is   expressed   analytically  thus :    <^  (a,   6)   is  a 

maximum  or  minimum  value  of  <^  (a;,  ^)  according  as  the 

increment 

^  (a  -f  A,  6  -f  A:)  —  (^  (a,  5) 

preserves  a  negative  or  a  positive  sign  for  all  values  of  the 
increments  A  and  h  which  are  numerically  less  than  a  given 
small  number  m. 

If  the  function  be  represented  by  the  ordinate  of  a  sur- 
face, then  a  maximum  (or  minimum)  ordinate  ^  (a,  6) 
is  greater  (or  less)  than  every  neighboring  ordinate 
^  (a  +  A,  h  -{■  It)  drawn  at  any  point  (a  +  A,  6  +  A;),  irre- 
spective of  the  signs  and  relative  magnitudes  of  A  and  h. 

107.  Determination  of  maxima  and  minima.  It  was  shown 
in  Art.  79  that  the  necessary  and  sufficient  condition  that  a 
function  of  one  variable  may  have  a  maximum  or  a  mini- 

183 


184  DIFFERENTIAL   CALCULUS  [Ch.  X. 

mum  for  a  given  value  of  the  variable  is  that  its  first  deriva- 
tive change  its  sign  as  the  variable  increases  through  the 
given  value.  Similarly  for  a  function  of  two  variables,  its 
differential  must  change  its  sign  at  a  maximum  or  minimum, 
independent  of  the  mode  of  variation  of  the  variables  through 
these  values.     Since 

dz  =  -^dx  -\ — —  dv, 
dx  dy 

and  since  either  x  ox  y  may  be  varied  alone,  the  first  neces- 
sary condition  is  that  the  coefficients  — ,  —  change  signs 

dx    dy 

separately ;  otherwise  it  would  be  possible  to  find  a  mode 
(or  direction)   of   variation   in  which  dz  does  not  change 

sign ;  for  instance,  if  —  does  not  change  sign,  then  dz  pre- 
serves its  sign  when  dy  is  zero  and  x  increases  through  a. 
Hence  the  critical  values  are  those  at  which 

^  =  0,    ^  =  0, 
dx  dy 

or  at  which  ^,  ^  become  infinite. 
dx    dy 

To  determine  whether  these  values  of  x^  y  will  give  a  maxi- 
mum or  minimum  value  to  z,  it  is  usually  impracticable  to 

test  the  signs  of  -^,  -^  for  all  neighboring  values  of  x,  y. 

dx     dy 
It  is  consequently  necessary  to  proceed  to  the  higher  deriva- 
tives.    Usually,  those  values  which  make  — ,  —  infinite, 

dx     dy 
will  also   make   successive   derivatives  infinite  ;   hence  sucli 

values  will  be  excluded  from  the  present  mode  of  investiga- 
tion. 

As  an  example  of  a  function  which   has  a  inininmm,  and  yet  has 
no  partial  derivatives,  consider 


107.]        MAXIMA  AND  MINIMA   IN   TWO   VARIABLES         185 

"When  X  =  0  and  y  =  0,  then  s  =  0;  but  for  every  other  value  of  x 
and  of  y,  z  must  be  positive ;  hence  z  =  0  is  a  niiuiinuni ;  but 

First  expand  the  f  imctioii  (})(a  +  h,  b  +  A:)  in  the  vicinity 
of  (a,  5)  by  Taylor's  theorem  ;  thus 

ax  ay 

2!(      dx^  dxdy  df) 

+  higher  powers  oi  h,  k; 

but.^=0,  ^=0;  hence 
dx  dy 

Criteria.    To  distinguish  between  a  maximum  and  a  mini- 
mum, at  both  of  which  -?=  0,  -^=0,  it   is  usually   suffi- 

dx  dy 

cient  to  consider  the  sign  of  the  expression  involving  terms 
of  the  second  degree  in  A,  k  ;  for  A,  k  can  generally  be  made 
so  small  that  this  expression  numerically  exceeds  tlie  sum  of 
all  the  subsequent  terms ;  hence  its  sign  will  determine 
the  sign  of  0  (a  -f  /t,  b  +  k')—  <^  (a,  6). 

When    x  =  a.    y  =  b,    let    ^=A,     -^=5,    ^  =  C, 

then  the  quadratic  expression  can  be  written  in  either  of 
the  forms 

1  (Ah?  +  2  Bhk  +Ck^-)  =  -L  (AVfi  -\-  2  ABJik  +  A  01^ 

The  first  term  of  the  numerator  of  the  last  form  is  always 
positive  or  zero ;    the  second  term  has  the  same   sign  as 


186  DIFFERENTIAL   CALCULUS  [Ch.  X. 

AG  —  W'.  If  the  latter  expression  is  positive,  the  numera- 
tor is  positive  for  all  values  of  A  and  k ;  but  if  it  is  negative, 
the  sign  of  the  fraction  will  depend  upon  the  values  of 
h  and  ^,  and  hence  there  can  be  no  maximum  nor  mini- 
mum ;  for  instance,  the  numerator  is  positive  when  A;  =  0, 
and  negative  when  A  and  k  are  so  taken  that  Ali-\-Bk  =  ^. 

The  second  indispensable  condition  for  a  maximum  or 
minimum  is,  therefore, 

B^<AO.  (1) 

This  being  satisfied,  the  numerator  is  positive,  and  hence 
the  sign  of  the  fraction  is  finally  determined  by  the  sign 
of  the  denominator  A.  If  A  is  positive,  ^  (a,  U)  is  a  mini- 
mum ;  if  A  is  negative,  </>(a,  F)  is  a  maximum. 

It  follows  from  the  condition  (1)  that,  since  &  is  posi- 
tive, A  and  C  must  have  the  same  sign. 

The  whole  process  may  be  summarized  as  follows :  to 
determine  whether  <^  (x,  y)  has  either  a  maximum  or  a 
minimum,  equate  its  first  partial  derivatives  to  zero,  and 

solve  the  resulting  equations  -2=0,  -^  =  0,  for  a;,  y.     Sub- 

dx  By 

stitute  these  critical  values  in  the  three  second  derivatives 

--^,  — 2_,  _^;  then  if  -— ^,  — ^  have  the  same  sign,  and 
OQT     dx  dy    ay^  ox^     oy^ 

\dx  By)       dx^     By^ 

there  is  a  maximum  when  the  common  sign  of  — ?  and  — ? 

^  B^  By^ 

is  negative,  and  a  minimum  when  it  is  positive. 

It  is  instructive  to  examine  the  form  of  the  representative 
surface  in  the  vicinity  of  the  critical  point,  especially  when 
some  of  the  conditions  for  a  complete  maximum  or  minimum 
are  not  satisfied.     The  geometric  meaning  of  all  the  condi- 


107]         MAXIMA  AND  MINIMA  IN   TWO   VARIABLES        187 

tions  (except  the  one  regarding  the  sign  of  B^—AC)  is 
immediately  evident  by  considering  the  conditions  that  the 
ordinate  may  have  a  turning  value  in  each  of  the  vertical 
sections  parallel  to  the  coordinate  planes.  The  deportment 
of  the  ordinate  in  the  intermediate  sections  depends  on  the 
sign  of  B^—AC,  the  discriminant  of  the  quadratic  expres- 
sion in  A,  k,  as  will  be  illustrated  in  the  examples. 

Special  cases  can  arise  in  which  A  =  0,  B=0,  (7=0,  or 
when  B^  —AC  =  0.  It  is  then  necessary  to  consider  the 
higher  degree  terms.  Instead,  however,  of  finding  general 
test  formulas  for  such  cases,  it  is  better  to  work  special 
examples  independently.  The  higher  degree  terms  can  in 
many  other  cases  be  made  to  give  useful  information  regard- 
ing the  deportment  of  the  function  in  the  vicinity  of  the  criti- 
cal value,  especially  in  cases  of  incomplete  maxima  or  minima. 
The  method  of  Art.  167  will  be  helpful  (Note  C,  p.  318). 

EXERCISES 

1.  Find  the  maxiinuin  and  minimum  values  of 

^  (a^»  y)  =  3  oj:^  -  x8  -  ?/8. 

Here  3^  =  3av-3x2;         ^  =  3ax-'dy^. 

dx  dy 

The  critical  values  are  therefore  x= a,  y=a;  x=0,  y=0. 

a!^=_6x;      i^=3a;      ^=-6y. 
dx^  dx  dy  dy^ 

At  a:  =  0,    y  =  0,     A  =0,    B  =  3a,     C  =  0, 
hence  (0,  0)  is  neither  a  maximum  nor  a  minimum. 
At  x  =  a,  y  =  a, 

A  =  -Qa,     B  =  3a,     C  =  -6a. 

In  this  case  both  r_r  and  ^-X  are  negative,  and  B^<  AC,  hence  <f>(a, a) 

has  a  maximum  value  a'. 

2.  Exhibit  graphically  the  deportment  of  the  function 

2  =  1  -  4  a:2  +  21  xy  -  5  .y^  \-  x^  +  y^ 
in  the  vicinity  of  the  critical  point  (0,  0). 


188 


DIFFERENTIAL   CALCULUS 


[Ch.  X. 


It  is  here  unnecessary  to  find  the  derivatives,  as  the  function  is  already 
expanded  in  the  vicinity  of  the  point  (0,  0),  the  letters  x  and  y  taking 
the  place  of  the  increments  h  and  k.  The  absence  of  the  first  degree 
terms  shows  that  the  point  (0,  0)  is  a  critical  point.  As  the  discriminant 
of  the  second  degree  terms,  21'^  —  4  •  4  •  5,  is  positive,  the  quadratic 
expression  has  real  factors,  and  can  therefore  be  made  to  change  its 
sign  for  different  ratios  of  y  to  x;  hence  there  is  no  complete  maximum 
nor  minimum. 

To  distinguish  the  sections  that  have  a  maximum  ordinate  at  this 
point,  from  those  that  have  a  minimum  ordinate,  write  the  equation  in 
the  form 

2  =  1  -  5  (  ?/  -  |]  (2^  -  4x)  +  x3  4-  2^8, 

which  shows  that  the  second  degree  expression  is  zero  when  the  ratio  of 
1/  to  a;  is  either  \  or  4 ;  positive  when  this  ratio  lies  between  \  and  4 ;  and 
negative,  for  all  other  values  of  the  ratio.     Hence,  all  vertical  sections 

V      1  y 

within  the  acute  angle  between  the  directions   -  =  -    and    -  =4   have  a 

°  X       o  X 

minimum  ordinate  at  (0,  0)  ;  and  all  vertical  sections  within  the  obtuse 
angle  have  a  maximum  ordinate  at  this  point.      In  the  first  transition 

direction  -  =  => 
X     a 


2  =  1    +x3+-^ 

125 


1    ,   126    a 

1  + x' ; 

125 


hence  the  increment  of  the  ordinate  is  positive  when  x  is  positive,  and 
negative  when  x  is  negative;  and  there  is  neither  a  maximum  nor  niini- 


FiG.  25. 


mum,  but  an  inflexion,  in  the  transition  section.     Similarly  for  the  other 
transition  direction.     The  two  horizontal  inflexional  tangents  in  these 


107.]        MAXIMA   AND  MINIMA   IN   TWO  VARIABLES        189 

vertical  sections  are  also  tangents  to  the  contour  of  the  section  made  by 
the  horizontal  tangent  plane  through  P  (Fig.  25). 

Some  idea  of  the  form  of  the  cubic  surface  at  the  critical  point  P  is 
given  in  the  figure.  It  shows  the  vertical  sections  XPX',  YPY'  in  the 
coordinate  planes,  in  both  of  which  OP  is  a  maximum;  the  transition 
sections  APA',  CPC\  the  contours  of  which  bend  upwards  in  the  first 
quarter,  and  downwards  in  the  third  quarter ;  and  an  intermediate  section 
BPB',  in  which  OP  is  a  minimum. 

If  the  third  degree  terms  were  absent,  the  transition  contours  APA', 
CPC  would  be  straight  lines,  the  surface  would  be  a  hyperbolic  parabo- 
loid, and  XYX'Y'  would  be  a  parallelogram. 

3.   Examine  the  deportment  of  the  function 

z  =  ~70  +  S8x-Q0y-  10x2  +  12xy  -  loy^ +  23^  -  y» 

in  the  vicinity  of  the  critical  point  (1,  —  2). 
Differentiation  and  substitution  give 

^=0,  ^=0,  ^  =  -8,  ^^=12,   ^  =  -18; 

dx  dy  3x2  QxQy  Qyi 

3!^=12.   -3^=0,  -^  =  0,5!^  =  -6. 

3X8  Qx'^Qy  QxQy-i  QyZ 

Hence  the  expansion  of  the  function  in  the  vicinity  of  the  point  (1,  —2)  is 

</.(!  +A,  -2  +  A;)=:«^(l,-2)-4A2  +  12Ayt-9ifc2-|-2A8-P. 

This  is  one  of  the  exceptional  cases  referred  to  above,  in  which  the 
discriminant  B^  —  AC  vanishes,  and  the  terms  of  the  second  degree  form 
a  complete  square.     Thus, 

</.(l  +h,  -2  +  /fc)-<^(l, -2)  =  -(2A-3;t)2  +  2A8-P, 

hence  the  increment  of  the  function  is  negative  for  all  small  values  of 
h  and  k,  unless  when  k  =  -^',  and  thus  the  ordinate  0(1,  —  2)  =  4  is  a 

maximum  in  every  vertical  section  but  one.  In  this  section  the  incre- 
ment of  the  function  is  A<^  =  2  A*  —  (§  K)^  =  If  A^,  hence  the  contour  of 
the  section  bends  upwards  in  the  first  quarter  and  downwards  in  the 
third  quarter. 

In  Fig.  26,  XPX'  and  YPY  are  the  contours  of  the  sections  parallel 
to  the  coordinate  planes,  and  APA'  is  the  contour  of  the  vertical  section 
in  the  intermediate  direction  k  =  \h.  This  may  be  regarded  as  a 
limiting  case  in  which  the  two  transition  directions  coincide.     The  hori- 


190 


DIFFERENTIAL   CALCULUS 


[Ch.  X. 


zontal  tangent  plane  at  P  cuts  the  surface  in  a  curve  which  has  a  cusp 
at  that  point;  the  cuspidal  tangent  coinciding  with  the  inflexional  tan- 
gent to  the  vertical  section  just  mentioned. 


Fio.  26. 

4.  Find  the  transition  directions  in  exercise  1  for  the  critical  point 
(0,  0),  and  show  the  form  of  the  surface  in  the  vicinity  of  the  point. 

5.  Examine  the  function  z  —  x'^  —  6  xy'^  +  ct/*  at  the  point  (0,  0). 
Show  that  if  c  >  9  there  is  a  minimum;  and  if  c  ^  9,  neither  maximum 
nor  minimum.     Draw  graph. 

6.  Show  that  0:6*+' """  has  neither  a  maximum  nor  a  minimum. 

7.  Divide  a  into  three  parts  such  that  their  continued  product  may  be 
a  maximum. 

8.  Find  the  minimum  surface  of  a  rectangular  parallelopiped  whose 
volume  is  a^. 

9.  What  value  of  x,  y  will   make    — — ^    a   maximum  or  a 

•   .  o  \  —  ax  —  by 

mmnnum  i  ^ 

10.  Find  the  values  of  x  and  y  that  make  sin  x  +  sin  y  +  cos  {x  +  y) 
a  maximum  or  a  minimum. 

11.  Find  the  maximum  of  (a  —  x)  (a  —  y)  (z  +  y  —  a). 

12.  The  electric  time  constant  of  a  cylindric  coil  of  wire  is 

_        mxyz 
ax  +  by  +  cz 

where  x  is  the  mean  radius,  y  is  the  difference  between  the  internal  and 
external  radii,  z  is  the  axial  length,  and  m,  n,  b.  c  are  known  constants. 
The  volume  of  the  coil  is  nxyz  =  g.  Find  the  values  of  x,  y,  z  to  make  m 
a  minimum  if  the  volume  of  the  coil  is  fixed.     (Perry's  Calculus.) 


107-108.]     MAXIMA   AND  MINIMA   IN   TWO  VARIABLES      191 

108.  Conditional  maxima  and  minima.  Maxima  and 
minima  of  implicit  functions.  In  certain  problems  the 
maximum  or  minimum  values  of  a  function  of  two  variables 
are  desired,  when  the  mode  of  variation  of  x  and  y  is  re- 
stricted by  an  imposed  condition. 

Let  the  function  be  z  =f(x,  y),  and  let  the  assigned  con- 
dition be  <^(x,  ^)  =  0  ;  then  it  is  required  to  find  the  maxi- 
mum or  minimum  values  passed  through  by  the  function  z, 
when  X  and  y  vary  consistently  with  the  relation  0  (x,  y)  =  0. 

This  problem  may  also  be  stated  in  the  following  geomet- 
rical form :  A  point  moves  on  the  surface  z  =f(x,  y)  in 
the  curve  of  intersection  made  by  the  cylindrical  surface 
^(x,  y)  =  0  ;  find  the  maximum  and  minimum  values  of  its 
height  above  the  horizontal  coordinate  plane. 

Since  the  variables  x  and  y  always  satisfy  ^(a:,  y)  =  0, 
hence  their  rates  of  change  are  connected  by  the  relation 

but,  since  z  is  at  a  turning  value,  its  rate  of  change  vanishes, 
hence 

therefore,  by  elimination  of  dx  and  dy^ 

^^_^^  =  0.  (3) 

dx  By       By  Bx 

This  equation,  together  with  <^(x,  y)  =  0,  will  determine 
the  critical  values  of  x  and  y. 

The  value  of  the  function  z,  corresponding  to  a  critical 
value,  will  be  a  maximum  or  minimum  according  as  d^z  is 
negative  or  positive  ;  but 

d:^=^Xda^+2^Ldxdy  +  ^.dy^  +  ^-f^^+^^^y^    (4) 
dx^  BxBy        '^       By^    ^       Bx  By    ^     ^  ^ 


192  DIFFERENTIAL   CALCULUS  [Ch.  X. 

to  eliminate  cPx,  d^i/,  multiply  (4)  by   -^  and  (5)  by  — , 

dt/  dy 

subtract,  and  take  account  of  (3)  ;  then 

+  (</)2/22-/2<^22)^^^^  (6) 

in  which  the  subscripts  1,  2,  indicate  differentiation  with 
regard  to  a;,  y^  respectively.  The  sign  of  the  right  hand 
member  of  (6)  is  not  changed  by  dividing  by  doc^^  and  then 

replacing   ^  by  —  t^,  from  (1);  hence  the  sign  of  dJ^z  at  a 

critical  point  is  the  same  as  the  sign  of 

T-  \.i'i>2fn-fl4>l\)4>i-  2(</>2/l2~/l</'l2)</>A 

+  (</>2/22-/2<^22)</'l^]   ; 

which  is  therefore  sufficient  to  discriminate  a  maximum 
from  a  minimum. 

Ex.  1.  If  z  =  x^-\-y^  —  ^axy,  and  if  x  and  y  vary  subject  to  the  condi- 
tion x2  +  ^2_  8a2.  show  that  z  passes  through  a  minimum  when  ar  =  2a, 
y  =  2a. 

Here,  /,  =  8(x2-fl?/),  f^=^y^-nx),  /ii  =  6ar,  f^^=-Za,  f^^^Qy, 
<f>,  =  2x,     <^2  =  2y,     «^„  =  2,     <f>,,^0,     «^22  =  2. 

The  critical  vahies  are  found  from  <^,/2  —  (^g/i  =  ^  and  x^  +  y'^  =  Sa^; 
and  one  pair  is  easily  found  to  be  ar  =  2  a,  y  =  2  a.     At  this  critical  point 
/i  =  6a2,    /2  =  6a2,    /j^  =  12a,    /j2=-3a,    /22=12a, 
<^^  =  4a,     <^2  =  4a,     <^ii=2,     <^i2  =  0,     ^22=2; 

and  the  sign  of  the  discriminating  expression  above  is  found  to  be  posi- 
tive, showing  that  z  is  a  minimum. 

Ex.  2.  Show  that  the  maximum  and  minimum  of  the  function  x^+y% 
subject  to  the  condition  ax^  +  2hxy  +  by^=l,  are  given  by  the  roots  of  the 
quadratic  equation 


108.]        MAXIMA   AND  MINIMA' IX  TWO   VARIABLES         193 

hence  show  how  to  find  the  axes  of  the  conic  defined  by  the  above 
equation  of  condition. 

Ex.  3'   Find  the  minimum  value  of  x'^  +  y'\  subject  to  the  condition 

-+f  =1. 
a     0 

Note.  When  the  equation  <^  (x,  y)  =  0  can  be  solved  for 
one  of  the  variables,  the  method  of  Art.  81  can  also  be  used. 

IMPLICIT   FUNCTIONS 

Let  y  be  defined  as  a  function  of  the  single  variable  x  by 
the  implicit  relation  /(a;,  y)  =  0 ;  it  is  required  to  find  at 
what  values  of  x  the  function  y  passes  through  a  maximum 
or  a  minimum. 

By  successive  differentiation,  leaving  the  independent 
variable  at  first  arbitrary, 

*"      5^       ^    dxdy        "     by"-  ^      bx  by    ^  ^^ 

by 
re,  y  at  whi 
one  of  the  equations 


hence  the  values  of  re,  y  at  which  — ^  changes  sign  satisfy 

ax 


bx  by 

Thus  the  first  set  of  critical  values  of  a;,  with  the  corre- 
sponding values  of  y,  are  to  be  found  from  the  simultaneous 
equations, 


194  DIFFERENTIAL   CALCULUS  [Ch.  X. 

and  the  second  set  of  critical  values  from 

/(^,y)  =  o,  1  =  0. 

These  two  sets  may  or  may  not  have  values  of  a;,  y  in 
common. 

Those  of  the  first  set  that  do  not  belong  to  the  second  set 

make  ^  =  0,  ^  =#=  0,  and  hence  make  -^  =  0. 
ox  dy  dx 

dv 
To  test  whether  -^  changes  its  sign,  in  passing  through 

zero,  the  method  of  Art.  82  is  available.     Taking  x  as  the 

independent  variable  in  (2)  and  putting  -^  =  0,  ^  =  0,  it 
gives 

^  =  -  ^. 

dx^  df ' 

Hence  for  the  critical  values  under  consideration,  y  is  a 

.  5¥    df 

maximum  or  a  minimum  according  as  ^■,  ~  have  the  same 

°        oar    oy 
or  opposite  signs. 

Those  of  the  second  set  of  critical  values  that  do  not  be- 
long to  the  first  set  make  -^  =  0,  ~  =^  0,  and  hence  make 
^  dy         '  dx 

-^  infinite. 
dx 

To  find  whether  -^  changes  its  sign,  the  second  derivative 
dx 

is  not  available,  since  it  and  all  subsequent  derivatives  are 
infinite ;  but  methods  of  trial  may  be  resorted  to,  in  which 
assistance  can  often  be  derived  from  the  graphical  represen- 
tation of  the  function. 

The  critical  values  that  are  common  to  the  first  and  the 

second  set  make  -^  =  0,  -^^  =  0,  and  hence  render  -^  inde- 
dx  dy  dx 


108.]       MAXIMA  AND  MINIMA   IN  TJVO   VARIABLES        195 

terminate  in  form.     When  numerically  evaluated  it  is  either 

zero,  infinite,  or  finite.     In  the  last  case  -^  cannot  change 

ax 

its  sign  and  there  is  no  turning  value  of  y.  In  the  first  two 
cases  the  question  whether  -^  changes  its  sign  as  x  passes 

through  the  critical  values,  and  ^  changes  correspondingly, 
is  to  be  decided  by  trial. 

Ex.  4.    Given      (x^  +  y^y  -2 i/(x^  +  y^)  -  x^  =  0; 

find  the  turning  values  of  y,  and  the  corresponding  values  of  x. 

o-  dy  2x  (x-  4-  f)  -  2  xy  -  x  ..^ 

dx  2  y  (x^  +  y^)  -  x^  -Sy^ 

the  fi.rst  set  of  critical  values  are  found  from 

(^■2  +  ^2)2  _.  2  y  (^jc-2  +  y^   -  X^  =  0,  (2) 

x[2(x2  +  y'0-23/-l]=0.  (3) 

Equation  (3)  is  satisfied  by  a:  =  0,  which,  substituted  in  (2),  gives 
y  =  0,  or  ^  =  2.     Equation  (3)  is  also  satisfied  when 

2(x2  + ^2) -22^-1  =  0, 

I.e.,  when  x^  =  —  y^-^  y  -k-\.,  which  substituted  in  (2)  gives  y  =  —  \,  whence 
z  =±  .43  •••.  Thus  the  first  set  of  critical  values  of  (x,  y)  is  composed 
of  the  four  pairs : 

(0,  0),     (0,  2),     (.43,   -  .25),     (-  .43,  -  .25). 

The  second  set  is  found  from  (2)  and  the  equation 

2K^2  +  2/2)-x2-3y2  =  o,  (4) 

which,  on  eliminating  x^,  gives  y  =  .75  or  0,  whence  x  =  ±  1.3  or  0. 
Thus  the  second  set  of  critical  values  of  x,  y  is  composed  of 

(0,  0),     (1.3,  .75),     (-1.3,  .75); 

the  values  (0,  0)  being  common  to  both  sets. 

To  test  the  remaining  critical  values  of  the  first  set,  use  the  second 
derivative  . 

(Py  -      ^-^^  -         Qx^  +  iy^-2y  -1 
dx^~      d£~     2y(x^  +  y^)  -x'^-S  y^ 

By 

DIFF.   CALC.  14 


196  DIFFERENTIAL   CALCULUS  [Ch.  X. 

which,  for  (0,  2)  is  negative,  and  for  (.43,  —.25),  (—.43,  —  .25)  posi- 
tive ;  hence,  when  x  passes  through  0,  the  function  y  passes  through  a 
maximum  value  2,  and  when  x  passes  through  —  .43,  .43,  y  passes 
through  a  minimum  value  —.25,  It  is  to  be  observed  that  in  the  latter 
case  the  function  has  three  other  values  (or  branches),  real  or  imagi- 
nary, that  do  not  pass  through  turning  values  when  x  passes  through 
±.43. 

To  test  the  critical  values  (0,  0),  for  which  equation  (1)  becomes  inde- 
terminate, evaluate  the  function  in  the  usual  way,  by  replacing  both 
numerator  and  denominator  by  their  respective  total  x-derivatives. 
This  gives 

(6x2  +  2  ?/2  _  2y  -  1) +  (4  x?/ -  2x)^ 
dy dx, 

^^  (ixy-2x)  +  (2x-^+Gy^-6y)^^ 

dx 

.-.  (2x^  +  Qy^  -  Q  y)(^'^y+  (8xy  -  ix)f'-^\  +  (6x^+2y^-2y-l)=0, 

a  quadratic  equation  in  -/•  Now  put  x  =  0,  y  =  0 ;  the  two  roots  of  the 
equation  become  infinite,  hence  -^=:cc.  In  the  present  case  it  is  easy  to 
find  by  trial  whether  -^  changes  sign ;  for  in  the  vicinity  of  the  values 
(0,  0)  equation  (1)  may  be  written  in  the  approximate  form 


dy  _      —x 
dx~  x^+3y^ 


41 


in  which  only  the  important  terms  are  retained ;  hence  -^  changes  sign 

dx 
from  +  to  —  as  a;  increases  through  zero,  and  thus  y  passes  through  a 
maximum. 

The  values  (0,  0)  could  also  be  shown  to  give  a  maximum  without  the 
use  of  derivatives,  by  observing  that  in  the  vicinity  of  th6  values  (0,  0) 
equation  (2)  can  be  replaced  by 

a;2-f2  2/8_o. 

When  X  is  small,  and  either  positive  or  negative,  y  must  be  negative ; 
but  when  x  =  0,  then  y  =  0  ;  hence  ^  =  0  is  a  maximum  value  of  the 
function. 

It  is  not  easy  to  test  the  other  critical  values  at  which  -^  becomes 

dx 
infinite  without  anticipating  the  methods  of  curve  tracing.    It  will  appear 
by  the  methods  of  Chapter  XVI II  that  the  graph  of  the  function  is  as 


108.]        MAXIMA  AND  MINIMA   IN    TWO  VARIABLES         197 

in  the  accompanying  figure,  and  that  the  critical  values  last  mentioned 
are  neither  tuaxiiua  nor  minima  values  for  y. 


Fig.  27. 

Ex.  5.  Given  x*  +  2  axhf  —  ay^  =  0 ;  find  the  maximum  and  minimum 
values  of  y,  and  of  x. 

Ex.  6.   li  x^  +  y^  —  3  axy  =  0 ;  find  the  maxima  and  minima  of  y. 

Ex.  7,   If  3  a^y^  +  xy^  +  4  cu^  —  0 ;  find  the  turning  values  of  x,  y. 

Ex.  8.  Show  that  in  the  vicinity  of  a  maximum  or  minimum  value 
of  /(x,  ?/),  the  increment  A/(x,  y)  is  an  infinitesimal  of  an  even  order, 
when  Ax  and  Ay  are  of  the  first  order. 

When  is  A/(x,  ?/)  of  the  third  order? 


CHAPTER   XI 


CHANGE  OF  THE  VARIABLE 


109.  Interchange  of  dependent  and  independent  variables. 
It  has  already  been  proved  in  Art.  22,  as  the  direct  conse- 
quence of  the  definition  of  a  derivative,  that  if  y=^(x), 
then 

dx      dx  (1) 

di/ 

This  process  is  known  as  changing  the  independent  varia- 
ble from  X  to  1/.  The  corresponding  relation  for  the  higher 
derivatives  is  less  simple,  and  will  now  be  developed. 


rr.                     d^y  .      ,              £    dx  dP'x 

lo  express   — ^  in  terms  oi    — ,  — — , 

aar                           dy  dy^ 
as  to  a;. 


differentiate   (1) 


d^y  _  d 
dx^     dx 


dx 
dy 


d_ 
dy 


but 


hence 


d^ 

dy 


1_ 

dx 


da? 


dx 

.dy\ 

df_^ 

dy) 

dy^ 
d^' 
dy) 
198 


dy 
dx 


dy 


1_ 

dx 
dy 


1_ 

dx 

dy 


(2) 


Ch.  XI.  109-111.]  CHANGE  OF   VARIABLE  199 

In  a  similar  manner, 

cPx  dx      ^f^x\^ 
^y  _      dfdy        \di/y 

\dy) 

110.   Change  of  the  dependent  variable.     If  y  is  o,  function 

of  z,  let  it  be  required  to  express  -f^,    ^^,  ...  in  terms  of 
J        »  dx     dx^ 

dz      d^z 

dx     dx^ 

Let  y  =  (f>(z^,  then 

dy^dydz_^  dz_^ 

dx     dz  dx  dx^ 

^y  _  d  fdy\  _  dz__d^f,,^  ^dz\  ^  dz 
dx^      dz\dxj    dx     dz\  dx)    dx 

but  the  second  term  can  be  expressed  directly  as  0'(z) 


dx^' 


hence 

g=  f  (.)(!)%  f«,^.  (4) 

The  higher  2:-derivatives  of  y  can  be  similarly  expressed 
in  terms  of  a:-derivatives  of  z. 

Ex.   Show  that  (4)  may  be  regarded  as  a  special  case  of  (6),  Art.  102, 
in  which  one  of  the  variables  is  replaced  by  a  constant. 

111.   Change    of    the    independent  variable.     Let  y  be  a 
function  of  x,  and  let  both  x  and  y  be  functions  of  a  new 

variable  t.     It  is  required  to  express  -^  in  terms  of  -^ ;  and 

— -^  in  terms  of  -^  and  — ^. 
dofi  dt  d^ 


200  DIFFERENTIAL   CALCULUS  [Ch.  XI, 

By  Arts.  21,  51, 

dy 

(1) 


(2) 


dy 
dy      dt 
dx ~  dx' 

dt 

d^y 

d^y  dx      d'^x  dy 
dt^  dt      dt^  dt 

dx^~ 

fdx\^ 

\dt) 

If  X  be  given  as  an  explicit  function  of  t  in  the  form 
=/(0,  then  - 
may  be  written 


finr  ft  'V 

x=f(t)^  then  —=f'(t^,  — -=/"(i),  and  the  last  equation 


dP^_dy^   f"it~) 
^y  _  W'      Tt  'fjt)  (3) 

d^~         [fit)f        • 

In  practical  examples  it  is  usually  better  to  work  by  the 
methods  here  illustrated  than  to  use  the  resulting  formulas. 

EXERCISES 

1.  Change  the  independent  variable  from  x  to  z  in  the  equation 

x^'^  +  x^  +  y^O,     whena:  =  e«. 
dx^        dx 

dx      dz 

dx^      dz^  dz 

Hence  a;2^  +  x'-^  +  w  =  0    becomes     ^^-  +  «  =  0. 

dx"^        dx     ^  dz^     ^ 

2.  Interchange  the  function  and  the  variable  in  the  equation 

dx^        ^\dxl 


111-112]  CHANGE  OF  VARIABLE  201 

3.  luterchange  x  and  y  in  the  equation 


R  = 


(Py 


4.  Change  the  independent  variable  from  j?  to  ^  in  the  equation 

\dx^l       dx  dx^     dx^\dxl 

5.  Change  the  dependent  variable  from  y  to  «  in  the  equation 

6.  Change  the  independent  variable  from  a;  to  y  in  the  equation 

x^  — -  +  x h  «  =  0,     when     y  =  log  x. 

dx^        dx 

7.  If  y  is  a  function  of  x,  and  x  a  function  of  the  time  t,  express  the 
//-acceleration  in  terms  of  the  x-acceleration,  and  the  x-velocity. 

Since  *  dy^^d^dx^ 

dt      dx  dt 

hence  ^  =  ^^  +  l^.l/^Y 

dfi      dx  dfi      dt      dtXdxl 

but  d_fdi\d_/dy\dx_d^dx^ 

dtXdxj     dxKdxj  dt      dx^  dt 

hence  ^^dr^d^^nlxy, 

di^      dxdt^     dx\dt) 

In  the  abbreviated  notation  for  ^-derivatives, 

d^y=^d^  +  fi(dxy. 
dx  dx^ 

Compare  this  result  with  (4),  Art.  110,  and  with  (6),  Art.  102. 

112.   Change  of  two  independent  variables.    Let  u  =f(x,  y') 
be  a  function  of  the  two  variables  2;,  y  which  are  themselves 

functions  of  two  new  variables  w,  2 ;  it  is  required  to  ex- 

du    du    •      ,  £   du     du 

press  — ,  —  in  terms  01  -— ,  --• 

dx    dy  ow    oz 


202  DIFFERENTIAL   CALCULUS  [Cii.  XI. 

I.    The  variables  x^  ^.explicit  functions  of  w,  z. 
Let  u  =f(x,  y);     x  =  <\>^(w,  z)\     ^  =  </)2(w,  2). 
Since  u  is  the  function  of  w  and  z, 

-— = -J- ~ -\- J-  -^     (2  regarded  as  constant). 
aw      dx  dw      By  dw 

The  values  of  — ,  -^  are  to  be  found  from  a;  =  <f>i,  y  =  4>a, 
dw    ow 

thus  du^dld^^df_d^ 

dw      dx   dw      dy  dw 


Similarly,  ^^f^Ax  +  f»^, 

dz      ox   oz       dy  oz    J 


(1) 

(w  regarded  as  constant). 
In  the  expression  for  t— »  2  is  to  be  regarded  as  constant, 
and  w  as  variable  ;  and  x,  y  as  functions  of  w. 

In  the  expression  for  — ,  w  is  to  be  regarded  as  constant, 

and  2  as  variable  ;  and  a;,  ^  as  functions  of  z. 

If  X,  y  be  called  the  old  variables,  and  z,  w  the  new 
variables,  then  it  appears  from  the  above  expressions  that 
when  the  old  variables  are  explicit  functions  of  the  new 

variables,  the  new  derivatives  -— ,  -r-  are  explicit  functions 

dw     dz 

07/         07/ 

of  the  old  derivatives  — ,  —    The  last  two  equations  may, 
dx    dy 

when  desired,  be  solved  for  — ,  —-• 

dx    dy 

II.    The  variables  tv,  2  explicit  functions  of  a;,  y. 

Let  z  =  y{r^(x,  y);     w  =  yjr^(x,  y}, 

,1                        du      du  dw  ,  du  dz      .  J  J  ,      ,N 

then  —  = 1 {y  regarded  as  constant), 

dx       dw  dx       dz  dx 

du       du  dw  ,   du  dz      .  ■,   j  ,      .  >, 

—  = 1 (x  regarded  as  constant). 

dy  .  dw  dy       dz  dy 


112.]                               CHANGE  OF   VARIABLE  203 

Substituting  the  values  of    — ,  — ,  — ,  —  from  z  =  >K» 

^                               dx     By     Bx    dy  ^' 

10  =  -sjr^,  the  last  equations  become 


dx  dw  dx  dz  dx  ' 
du  _  du  dyfr^  du  dyjr.^ 
dy      dw  dy        dz    dy 


(2) 

These  equations  may,  when  desired,  be  solved  for  — ,  — . 

dz     dw 

In  this  case  the  new  variables  are  explicit  functions  of 

the  old  ones,  while  the  old  derivatives  are  explicit  functions 

of  the  new  ones. 

Ex.    Let   u  =  x^  —  y^,   x  =  pcosO,   y  =  psinO.     Find  2_,  ?L    by  the 
method  of  I.  ^P   ^^ 

dn^dudx^dudl     (^  regarded  as  constant), 
dp     dx  dp     dy  dp 

=  2x  cos  0  —  2y  sin  9, 

=  2pcos2e-2psin2^, 

=  2pcos2d, 

which  agrees  with  the  result  of  direct  substitution. 

*«*'"'     % = 1 1 + in  <"  ''^^'^ "  "°""'^"'>' 

=  —  2  xp  sin  6  —  2yp  cos  $, 

=  —  4  p2  cos  6  sin  6, 

=  -2p^8m26, 

which  also  agrees  with  the  result  of  direct  substitution. 

Next  suppose  the  new  variables  p,  6  are  expressed  in  terms  of  the  old 

variables  x,  y  in  the  form  p  =  Vx^  +  y%  6  =  tan-'^  ;  find  ^,  ^  by  the 
method  of  II.  " 

Here       ^=        ^        =  cos  g;        ^^  =  _:li^  =  -  5111^ 
dx      Var2  +  y2  dx      x^  +  y^  p 

d£=        y        =sin6i         ^=    -^-  =  22!i^ 
dy      Vx2  +  yi  dy     x^  +  y'        p 


204  DIFFERENTIAL   CALCULUS  [Ch.  XI. 

but  ^  =  2  X  =  ^  ^  +  ^  1^  (y  regarded  as  constant), 

ox  dp  ax      da  ox 

u  o  ^      6"        zi     5"     sin  ^  ,,v 

hence  2  p  cos  0  ~^--  cos  &  —  ^  • ;  (1) 

dp  da       p 

also,  ^  =  -2^/=^^  +  ^"^  (X  regarded  as  constant), 

ay  dp  dy     off  dy 

hence  _  2psin5  =  ^  sin^  +  ^  2^.  (2) 

'^  dp  dd     p  ^  ^ 

Now,  solving  (1)  and  (2)  for  ^,  ^,  it  follows  that 

dp    ou 

^  =  2pcos2e,      g  =  -2psin2d; 
op  da 

the  same  results  as  were  obtained  before. 

III.  The  relation  between  a;,  y  and  w,  z  defined  by  implicit 
equations. 

Let  fi(x,  y,  z,  w)  =  0,       f^(x,  y,  z,  w)  =  0. 

In  the  first  place,  to  find  — ,  -^  required  in  I  (1),  differ- 

oz    dz 

entiate  the  two  given  equations  partially  with  regard  to  2, 
then 

dz       dx  dz      dy  dz 

(w  regarded  as  constant) 

dz       dx  dz      dy  dz 

solve  the  resulting  equations  for  — ,  -^,  then  substitute  in 
(1)  and  proceed  as  before. 

bimilarly  tor  — ,  -^• 
dw   dw 

113.  Change  of  three  independent  variables.  The  student 
will  not  have  much  difftculty  in  extending  the  theory  to 
functions  of  three  or  more  variables. 


112-114.]  CHANGE  OF  VARIABLE  205 

Let  u  =/(.r,  y,  z')  ;  and  let  x,  y,  z  be  functions  of  three 
new  variables  w,  v,  w,  connected  by  the  equations 

X  =  cf)^(u,  V,  w),  y  =  <^2(w,  w,  w),  z  =  <^3(w,  V,  M>).       (1) 
It  is  required  to  express  -^  in  terms  of  w,  v,  w. 

^=^-^  +  -^-^4-^--  (w,  w  regarded  as  constants), 
dw      da:  dw      oy  du      az  au 

df       df  dx       df  dy       df  dz  ^  ,    ,  ^      ^  -v 

~  =  -^^--h^-:^  +  ~^  (m,  w  regarded  as  constants), 
dv       ox  dv      ay  av       az  du 

—  =  —  — .-f-^_^4--Z  —  (m,  V  regarded  as  constants). 
aw      ax  aw      dy  div      dz  dw 

finr       fi'T*      n'y*       fiti 

From  (1),  — ,    — ,    — ,    -^,  •••  can  be  found  ;  their  values 
du     dv     div     du  ^j. 

are  to  be  substituted  in  the  equations  for  ~i  ••*,  and  the  re- 

df     df 
suiting  equations  solved  for  -Z.,    -ii.,  ..-. 

dx     dy 

Similarly  for  the  case  in  which  u,  v,  w  are  explicit  func- 
tions of  X,  y^  z. 

114.  Application  to  higher  derivatives.  The  second  and 
higher  derivatives  can  be  obtained  in  the  same  way.  As 
the  general  formulas  become  too  complicated  to  be  of  much 
use,  it  is  better  to  work  out  special  examples  independently. 

Ex.     Express  2^  +  2_H  in  terms  of  p,  $,  given 
dx^      dy^ 

X  ■=  p  cos  6,  y  =  p  sin  6.  (1) 

The  general  formula  is 

du  _  du  d£_  ,  d^  dO^ .  /2) 

dx      dp  dx      dd  dx ' 

in  which    ^,    —   are  to  be  obtained  from  (1),  by  differentiating  and 

dx    dx 
solving. 


206  DIFFERENTIAL   CALCULUS  C^h.  XI. 

Thus  l=^cosd-psme^, 

dx  dx 

{y  regarded  as  constant)  ; 

0=^sine  +  pCQse^A 
dx  Qx 

hence  QR  =  ^os  0,  ^  = -^^-  '  (3) 

dx  dx  p  ^  ^ 

Similarly    ^,    £^  can  be  obtained  from  (1)  : 

^   dy    dy  ^  ' 


hence 


O  =  ^cose-psin^^, 

dy           ^        dy 

(x  regarded  as  constant) ; 

dy            ^         dy 

5fi  =  sine,    Se^cosd. 
dy                dy        p 

(4) 

Substitution  from  (3)  in  (2)  gives 

3^  =  ^cos^-^^.  (5) 

dx     dp  dO     p  ^  ^ 

A  repetition  of  this  process  gives 

^-^    Q  2g         dhi  sing  cos  g      dusm^O       d^   cos  g  sing 
dx^     dp^  dO  dp        p  dp     p         dO  dp        p 

d^u  sin^  6      du  cos  6  sin  6  .  du  sin  6  cos  6  x«v 

d6'    p''       dd       p^  dO       p^      '  ^  ^ 

The  expression,  similar  to  (2),  for    — ,  combined  with  (4),  leads  to 

dy 

^  =  dn  ^^      ducose  7 

dy     dp  dO    p   '  ^  ^ 

and  when  this  step  is  repeated,  there  results, 

a^w      a^u    .  „/)  ,     d'^u   sing  cos  g  ,  ^ucos^g  ,     d^u   cos  g  sing 
a.y^     dp^  dOdp        p  dp     p         da  dp        p 

d^u  cos^g      5"  cos  gsin  g      ^m  sin  gcosg.  ^gv 

ag-'^^      go     -2        ^     -2     '  ^> 

and  the  addition  of  (G)  and  (8)  gives  the  required  identity 
d^      d^u  ^d^u      1  du  .  1   d^u 

dx^    dy""    dp^    pdp    p^dB^' 


114.]  CHANGE  OF  VARIABLE  207 

EXERCISES 

1.  Given  x  =  p  cos  0,  y  =  p  sin  B,  y  being  a  function  of  x,  show  that 

d^'~  (^cose'l^-psmey 

2.  Given  x  =  a(l  —  cos  t),  y  =  a(nt  +  sin  t);  prove  that 

d'hf  _  _  n  cos  t  +  1 
dx'^  a  sin^  t 

3.  If  $  =  X  cos  a  —  y  sin  a,   rj  =  x  sin  a  +  y  cos  a,  prove  that 

4.  Given  x  =  p  cos  $,  y  =  p  sin  ^,  show  that 

dx     ^  dy     dd         dx     ^  dy     ^  dp 

5.  If  x  =  p  cos  0,  y  =  p  sin  6,  show  that  the  expression 


^  ^   [del     ^d&^ 


APPLICATIONS   TO    GEOMETRY 


CHAPTER   XII 
TANGENTS  AND  NORMALS 

115.    It  was  shown  in  Art.  17  that  if  f(x,t/')  =  0  be  the 

equation  of  a  plane  curve,  then   -^  measures  the  slope  of 

ax 

the  tangent  to  the  curve  at  the  point  x,  y.     The  slope  at  a 

particular  point   (x^^y^  will  be  denoted  by  -^,  meaning 

dx^  a/ 

that  x^  is  to  be  substituted  for  x.  and  y,  for  «  in    -^  = ;. 

^  ^^        ^        dx         QJ^ 

after  the  differentiation  has  been  performed.  '  3« 

116-    Equation  of  tangent  and  normal  at  a  given  point. 
Since  the  tangent  line  goes  through  the  given  point  (a^j,  y-^ 

and  has  the  slope  -O,  its  equation  is 
dx-^ 

y-y\=^i^-^iy  (1) 

The  normal  to  the  curve  at  the  point  (ajj,  y{)  is  the  straight 
line  through  this  point,  perpendicular  to  the  tangent. 

208 


Ch.  XII.  115-117.]     TANGENTS  AND  NORMALS 


209 


t.e« 


dx       1 
Its  equation  is,  since  -_.=__  bv  Art.  22, 
dy     dy     ^ 

dx 
dx^  .  ,, 


(2) 


117.    Length  of  tangent,  normal,  subtangent,  subnormal. 

The  portion  of  the  tangent  and  normal  intercepted  between 
the  point  of  tangency  and  the  axis  OX  are  called,  respec- 
tively, the  tangent  length  and  the  normal  length;  and  their 
projections  on  OX  are  called  the  subtangent  and  the  sub- 
normal. 


Pio.  28  a. 


Fig.  28  &. 


Thus,  in  Fig.  28,  let  the  tangent  and  normal  at  P  to  the 
curve  PC  meet  the  axis  OX  in  ^  and  N,  and  let  MP  be  the 
ordinate  of  P,  then  TP  is  the  tangent  length, 

PN  the  normal  length, 

TM  the  subtangent, 

MN  the  subnormal, 

which  will  be  denoted,  respectively,  by  ^,  w,  t,  v. 
Let  the  angle  XTP  be  ^,  then  tan  <f>  =  m,  say  : 

1 


cos  <f>  = 


ViT 


sin^  = 


m 


W 


VIT 


w 


210  DIFFERENTIAL   CALCULUS  [Ch.  XII. 

V^         Vt  VI  +  m^  y,  ,-. s 

t  =  -^  =  ^i ;    n  =  -^=v,Vi  +  m^; 

sin  9  m  cos  (f> 

.  ,  dx-,       Vi  ^       ,  duy 

T  =  y,cot<^  =  y,^  =  ^;    v  =  y,i'^n4>  =  y,-£^=my,. 

The  subtangent  is  measured  from  the  intersection  of  the 
tangent  to  the  foot  of  the  ordinate ;  it  is  therefore  positive 
when  the  foot  of  the  ordinate  is  to  the  right  of  tlie  intersec- 
tion of  tangent.  The  subnormal  is  measured  from  the  foot 
of  the  ordinate  to  the  intersection  of  normal,  and  is  posi- 
tive when  the  normal  cuts  OX  to  the  right  of  the  foot  of 
the  ordinate.  Both  are  therefore  positive  or  negative,  ac- 
cording as  ^  is  acute  or  obtuse. 

The  expressions  for  t,  v  may  also  be  obtained  by  finding 
from  equations  (1),  (2),  Art.  116,  the  intercepts  made  by 
the  tangent  and  normal  on  the  axis  OX.  The  intercept  of 
the  tangent  subtracted  from  x^  gives  t,  and  a^j  subtracted 
from  the  intercept  of  the  normal  gives  v. 

EXERCISES 

1.  In  the  curve  y  (a:  —  1)  (x  —  2)  =  x  —  3,  show  that  the  tangent  is 
parallel  to  the  axis  of  x  at  the  points  for  which  x  =  3  ±  V2.     ^ 

2.  Write  down  the  equations  of  the  tangents  and  normals  to  the 

curve  y  =  —^ —  at  the  points  for  which  ?/  =  -• 
a^  +  x^  '4 

3.  Find  the  equations  of  the  tangents  and  normals  at  the  point  (xj,  y{) 
on  each  of  the  following  curves : 

(a)  x^  +  y'^  =  c\  (e)  xy  (x  +  ?/)  =  a\ 

(i)   xy  =  k"^,  (d)  e"  =  sin  x. 

4.  Prove  that  — |-  -  =  1  touches  the  curve  y  =  be  "  at  the  point  in 

a      b 
which  the  latter  crosses  the  axis  of  y. 

5.  Find  the  points  on  the  curve 

2/=:(x-l)(x-2)(x-3) 
at  which  the  tangent  is  parallel  to  the  axis  of  x. 


117.]  TANGENTS  AND  NORMALS  211 

6.  Find  the  intercepts  made  upon  the  axes  by  the  tangent  at  (Xj,  y{) 
to  the  curve  Vx  +  y/i/  =  Va,  and  show  that  their  sum  is  constant. 

7.  In  the  curve  x^y'^  —  a^(x  +  //),  the  tangent  at  the  origin  is  inclined 
at  an  angle  of  135°  to  the  axis  of  x. 

8.  In  the  curve  x^  +  ys  =  a^,  find  the  length  of  the  perpendicular 
from  the  origin  on  the  tangent  at  (xj,  ?/j)  ;  and  the  lengtli  of  that  part  of 
the  tangent  which  is  intercepted  between  the  two  axes.     (A.  G.,  p.  323.) 

9.  Show  that  all  the  curves  represented  by  the  equation 

when  different  values  are  given  to  n,  touch  each  other  at  the  point  (a,  b). 

10.  Show  that  all  the  points  of  the  curve 

y^  =  4:  a  Ix  +  a  sin  -\, 
at  which  the  tangent  is  parallel  to  the  axis  OX  lie  on  a  certain  parabola. 

11.  Prove  that  the  parabola  y^  =  i  ax  has  a  constant  subnormal. 

12.  Prove  that  the  circle  x^  +  y"^  =  a^  has  a  constant  normal. 

13.  Show  that  in  the  tractrix,  the  length  of  the  tangent  is  constant ; 
the  equation  of  the  tractrix  being 

X  =  Vc^  —  y^  +  -  log ^  • 

2    "c  +  y/^nryi 

X 

14.  Show  that  the  exponential  curve  y  =  ae'  has  a  constant  subtan- 
gent. 

15.  At  what  angle  does  the  circle  x^  +  y^  =  Sax  intersect  the  cissoid 

y2  =  ^!_?     (A.  G.,  p.  309.) 
2a  —  X 

x» 


16.   Find  the  subtangent  of  the  cissoid  y^  = 


2a  -X 


17.  Find  the  normal  length  of  the  catenary  y  =  ^(f'"  +  e  »). 

18.  Show  that  the  only  Cartesian  (x,  y)  curve  in  which  the  ratio 
of  the  subtangent  to  the  subnormal  is  constant  is  a  straight  line. 

19.  Show  that  the  equation  of  the  tangent  to  the  curve  f(x,  y)  =  0 
at  the  point  (x,,  ^j)  may  be  written 

DIFF.  CALC.  —  15 


212 


DIFFERENTIAL   CALCULUS 


[Vh.  Xil. 


20.     Prove  that  the  equation  of  the  tangent  to  the  curve 
z^  —  3  axy  +  y^  z=  0 
may  be  written  Xj^  —  ax^y  —  axy^  +  y-^y  =  ax^y^. 


POLAR  COORDINATES 
118.  When  the  equation  of  a  curve  is  expressed  in  polar 
coordinates,  the  vectorial  angle  0  is  usually  regarded  as  the 
independent  variable.  To  determine  the  direction  of  the 
curve  at  any  point,  it  is  most  convenient  to  express  the  angle 
between  the  tangent  and  the  radius  vector  to  the  point  of 
tangency. 

Let  P,  Q  be  two  points  on  the 
curve  (Fig.  29).  Join  P,  Q  with 
the  pole  0,  and  drop  a  perpen- 
dicular PMirom  P  on  OQ.  Let  /a, 
0  be  the  coordinates  of  P  ;  p  -\-  Ap, 
6  +  Ad  those  of  Q  ;  then  the  angle 
POQ=Ae;  PM=p  sin  A0;  and 
MQ=OQ-OM=p  +  Ap-p  cos  A0; 

p  sin  A^ 


F««.  29. 


hence 


tan  MQP  = 


p  -\-  Ap  —  p  cos  A0 

When  Q  moves  to  coincidence  with  P,  the  angle  31 QP 
approaches  as  a  limit  the  angle  between  the  radius  vector 
and  the  tangent  line  at  the  point  P.  This  angle  will  be 
designated  by  yfr. 

p  sin  A^ 


Thus 


but 


hence 


p(l  -  cos  Ae)=2p  siu2  1-  A^, 

p  sin  A0 
Ad 


t&nyir  =  ^^2o 


•     1  \a    sin  1 A^  ,  Ap' 


117-120.] 

but 

Therefore 


TANGENTS  AND   NORMALS 


213 


lini     sill  A^  _  . 

^6  =  0—^0--^^ 


lira    sin  1  A0  _  J 


A^  =  0 


hM 


,        .       p         dd 
dd 


Examples  on  dynamical  interpretation. 

Ex.  1.  A  point  describes  a  circle  of 
radius  p;  prove  that  at  any  instant  the 
arc  velocity  is  p  times  the  angle  velocity; 


(Is 


fid 


Ex.  2.    When    a    point    describes    any 
curve,  prove   that   at   any  instant  the  ve- 
locity —  has  a  radius  component  ^  and 
^  dt  ^  (It 

a  circle  component  p  — ,  and   hence  that 
dt 

I       dp      •     ,         dd    ,        ,         dd 

cosi/f  =  -^,  sin  ip  =  p — ,  tan  tp  —  p — 

ds  ds  dp 


Fm.  31. 


119 


dy 


pd9 


Relation  between  ^  and  ^-.     If 
ax  dp 


the  initial  line  be  taken  as  the  axis  of  z, 
the  tangent  line  at  P  makes  an  angle  (f> 
with  this  line  by  Art.  117.     Hence 

e  +  ylr  =  <t>; 


FiQ.  82> 


120.  Length  of  tangent,  normal,  polar  subtangent,  and  polar 
subnormal.  The  portions  of  the  tangent  and  normal  inter- 
cepted between  the  point  of  tangency  P  and  the  line  through 
the  pole  perpendicular  to  the  radius  vector  OP  to  the  point 
of  tangency,  are  called  the  polar  tangent  length  and  the  polar 


214 


DIFFERENTIAL   CALCULUS 


[Ch.  XII. 


normal  length ;  and  their  projections  on  this  perpendicular 
are  called  the  polar  subtangent  and  polar  subnormal. 


Fio.  33  &. 


Fig.  33a. 

Thus,  let  the  tangent  and  normal  at  P  meet  the  perpendic- 
ular to  OP  in  the  points  iV,  M.     Then 

PN  is  the  polar  tangent  length, 
PM  is  the  polar  normal  length, 
OiV  is  the  polar  subtangent, 
OM  is  the  polar  subnormal. 

They  are  all  seen  to  be  independent  of  the  position  of  the 
initial  line.  The  lengths  of  these  lines  will  now  be  consid- 
ered. 

Since  PN=  OP  sec  OPN=  p  sec 


t-VKIF 


'-(!)• 


hence     polar  tangent  length  =  p—y^p^ '^{^i 


d0 


Again,  OiV=  OP  tan  OPN=  p  tan  f  =  p^"^ 

hence  polar  subtangent  =  p^  —  . 

dp 


120.]  TANGENTS  AND  NORMALS  215 

PM  =  OF  CSC  OFN  =pcsc^|r=y|f^  +  f^Y, 


Ud. 

hence        polar  normal  length  =  "Y/a^  -]-(-£]. 

0M=  OP  cot  OPN  =  ^'> 
pdu 

hence  polar  subnormal  =  -^• 

ad 

The  signs  of  the  polar  tangent  length  and  polar  normal 
length  are  ambiguous  on  account  of  the  radical.  The  di- 
rection of   the  subtaugent   is  determined   by   the   sign   of 

p2 —  :  when  —  is  positive,  the  distance  ON  should  be  meas- 
dp  dp 

ured   to  the  right,   and    when    negative,   to  the  left  of   an 
observer  placed  at   0  and  looking  along   OP;    for  when  6 

increases  with  o,  -—  is  positive  (Art.  20),  and  -^  is  an  acute 
dp  ^0 

angle  (as  in  Fig.  33  6) ;  when  6  decreases  as  p  increases,  — 
is  negative,  and  -^  is  obtuse  (Fig.  33  a). 


EXERCISES 

1.  Show  that  the  polar  subtangent  is  constant  in  the  curve  pB  =  a. 

2.  Show  that  in  the  curve  p  =  a-  e*  =<"<»,  the  tangent  makes  a  constant 
angle  a  with  the  radius  vector.  For  this  reason,  this  curve  is  called  the 
equiangular  spiral.     (A.  G.,  p.  330.) 

3.  For  the  same  curve  as  in  Ex.  2,  find  the  polar  subtangent  and  polar 
subnormal. 

4.  Find  the  angle  of  intersection  of  the  curves 

p  =  a  (1  +  cos  d),    p  =  ft  (1  -  cos  6}. 

5.  In  the  circle  p  =  a  sin  6,  find  if/  ajid  <f>. 

6.  In  the  curve  p  =  a$,  show  that  tan  ij/  =  B,  and  that  the  polar  sub- 
normal is  constant.     (A.  G.,  p.  325.) 

a 

7.  In  the  parabola  p  =  a  sec^  -,  show  that  <f>  +  ij/  =  ir. 


CHAPTER  XIII 

DERIVATIVE  OF   AN  ARC,   AREA,  VOLUME    AND   SURFACE 
OF  REVOLUTION 

121.  Derivative  of  an  arc.     The  length  s  of  the  arc  AP 

of  a  given  curve  ^  =/(a;),  measured  from  a  fixed  point  A 
to  any  point  P,  is  a  function  of  the  abscissa  x  of  the  latter 
point,  and  may  be  expressed  by  a  relation  of  the  form 
8  =  <j>(x). 

The  determination  of  the  function  ^  when  the  form  of 
/  is  known,  is  an  important  and  sometimes  difficult  problem 

in  the  Integral  Calculus.     The  first  step  in  its  solution  is 

ds 
to  determine  the  form  of  the  derivative  function  — -  =  <f>'(jt>)'> 

ax 

which  is  easily  done  by  the  methods  of  the  Differential  Cal- 
culus. 

Let  P^  be  two  points  on  the  curve  (Fig  34);  let  x,  y 

be  the  coordinates  of  P;  x  -\-  Aa;, 
y  +  Ly  those  of  ^ ;  s  the  length 
of  the  arc  AP ;  s  +  As  that  of 
th«  arc  AQ.  Draw  the  ordinates 
MP^  NQ ;  and  draw  PR  parallel 
to  MN\  tlien  PR  =  Aa:,  RQ  =  Ay; 
arc  PQ  =  As.     Hence 


Y 

A      J 

A 

/ 

R 

X 

0 

A 
Fio 

1 1 

84. 

V 

Chord  PQ  =  V(Aa;)2+(Ay)2, 
216 


Ch.  XIII.  121-122.]      DERIVATIVES  OF  ARC,   ABEA,  ETC.     217 


T-e-!-:.=li-f-5=||V^^- 


Ax' 

Taking  the  limit  of  both  members  as  Ax  =  0  and  putting 

^xto^=  1'  ^y  A^^-  1^'  Th.  4,  and  Art.  10,  Th.  10,  Cor., 

it  follows  that 

ds 
dx 


Similarly 
and 


I.e. 


(1) 

(2) 

(3) 
(4) 


122.  Trigonometric  meaning  of  |^.»  ~- 


Since  -—  = 


Ax 
As 


— --  .  — ^  =  cos  RPQ  •  — ~i 
PQ     As  ^     As 


it  follows,  by  taking  the  limit,  as  Ax  =  0,  that 

dx 


ds 


=  cos  <f>. 


wherein  <^,  being  the  limit  of  the  angle  RPQ,  is  the  angle 
which  the  tangent  drawn  at  the  point  (x,  y)  makes  with  the 
a;-axis. 

Similarly,  -^  =  sin  ^ ;  whence  —  =  sec  </> ;  —  =  esc  <^. 


ds  "^  '  dx 

Using  the  idea  of  a  rate  or  dif- 
ferential, all  these  relations  may 
be  conveniently  exhibited  by  Fig. 
35. 

These  results  may  also  be  de- 
rived from  equations  (1),  (2)  of 

dy 
Art.  121,  by  putting  ^  =  tan  ^. 


dy 


<^y. 


ds 
dx 


Fis.85. 


218 


DIFFERENTIAL   CALCULUS 


[Ch.  XIII. 


123.  Derivative  of  the  volume  of  a  solid  of  revolution. 
Let  the  curve  APQ  revolve  about  the  a;-axis,  and  thus  gen- 
erate a  surface  of  revolution ;  let  V  be  the  volume  included 
between  this  surface,  the  fixed  initial  plane  face  generated 
by  the  ordinate  AB^  and  the  terminal  face  generated  by  any 
ordinate  MP. 

Let  A  P^  be  the  volume  generated  by  the  area  PMNQ ; 
then  A  V  lies  between  the  volumes  of  the  cylinders  gener- 
ated by  the  rectangles  PMNR  and  SMNQ ;   that  is, 

iry^i^x  <  AV  <  7r{i/  +  Ai/yAx. 
Dividing  by  Ax  and  taking  limits, 


124.  Derivative  of  a  surface  of  revolution.     Let  *S'  be  the 

area  of  the  surface  generated  by  the  arc  AP  (Fig.  36);  and 

A*S'  that  by  the  arc  PQ,  whose  length  is  As. 

Draw  PQ',  QP'  parallel  to  OX 
and  equal  in  length  to  the  arc 
PQ;  then  it  may  be  assumed  as 
an  axiom  that  the  area  generated 
by  P^  lies  between  the  areas  gen- 
erated by  PQ'  and  P'Q;  i.e., 

2  TryAs  <  AS  <  2  7r(?/  +  Ai/')As. 

Dividing  by  As  and  passing  to  the  limit, 

dS 


ds 


=  2  Try, 


dS^dSds^^      J^fdy}?, 

rint*  /7o  rinr  '  V  /I ^^  f 


dx      ds     dx 


\dxJ 


(1) 
(2) 


123-125.]  DERIVATIVES  OF  ARC,  AREA,   ETC.  219 

125.   Derivative  of  arc  in  polar  coordinates. 

Let  p,  6  be  the  coordinates  of  P ;  p  +  Ap,  0  +  Ad  those 
of  ^  ;  «  the  length  of  the  are  KP  ; 
A«  that  of  arc  PQ.     Let  PM  be 
perpendicular  to  OQ  ;   then 


P3I=  psmA0, 

3IQ  =  OQ-OM=p+Ap-p  cos  Ae 
=  p(l  —  cos  A^)  +  A/>  o^ 

=  2psin2iA^  +  A/o.  ^^•"• 

Hence        P  ^  =  (/a  sin  A(9)2  -f  (2  p  sin^  i  A^  +  A/))2, 

/PO      A.9V 

Replacing  the  first  member  by  [  — —  •  — ^  1 ,  passing  to  the 

limit  when  A^  =  0,  and  putting  lim  — -  =  1,  lim ^r—  =  1, 

siniA6'  ^         ^  As        '  A^  ' 

lim  '2        =  1   it  follows  that 
^A^  ' 


I.e., 


l=v^HI)- 


In  the  rate  or  differential  notation  this  relation  may  be 
conveniently  written 

and  its  dynamic  interpretation  is  shown  in  the  figure  of 
Art.  118  (Fig.  31). 


220  DIFFERENTIAL   CALCULUS         [Ch.  XIII.  126. 

126.  Derivative  oi  area  in  polar  coordinates.      Let  A  be 

the  area  of  OKP  measured 
from  a  fixed  radius  vector  OK 
to  any  other  radius  vector  OP; 
let  A-4  be  the  area  of  OPQ. 
Draw  arcs  PM,  QN,  with  0  as 
a  center;  then  the  area  POQ 
lies  between  the  areas  of  the 
^•''•''-  sectors  0PM  and  ONQ;   i.e., 

1  ffi^e  <^A<  ^  (p  +  A/j)^  as. 

Dividing  by  A0  and  passing  to  the  limit,  when  A0  =  0,  it 

follows  that 

dA 
dd 


^=iv. 


For  the  derivative  of  the  area  of  a  curve  in  x,  y  coordi- 

dA 
nates,  see  Art.  17.     The  result  is  — -  =  v. 

dx 


EXERCISES 

1.  Given  ?-:  +  ^^=l;  find  ^,^,^^ 

a^      h^  dx    dx    dx     dx 

2.  Similarly  for  the  parabola  ?/^  =  4  ax. 

3.  In  the  curve  e'Ce'  -  1)  =  e*  +  1,  show  that  —  =  ^^  "^    • 

4.  If  <f)  be  the  eccentric  angle  of  the  ellipse  —  +  ^  =  1,  prove  that 
ds  , .  a2     62 

~  =  ovl  —  e^  cos^  <f>,     e  being  the  eccentricity. 
d<p 

[dx  —  —  a  sin  <fyd<f>f  dy  =  b  cos  <f>d<^,  ds"  —  (a"'^  sin''^  <f>  +  b^  cos^  <f))d<f>'^,  etc.] 

5.  Given  p  =  a  cos  6 ;  find  ^—-,  — -• 

■  dd    dd 

6.  In  p2  =  a^cos  2  6,  show  that  ^  =  — 

dd      p 

as 


7.   Given  p  =  a  (1  +  cos  6),  prove  ^=  ■\/2ap. 


CHAPTER  XIV 
ASYMPTOTES 

127.  When  a  curve  has  a  branch  extending  to  infinity,  the 
tangents  drawn  at  successive  points  of  this  branch  may  tend 
to  coincide  with  a  definite  fixed  line  as  in  the  familiar  case 
of  the  hyperbola ;  or,  on  the  other  hand,  the  successive 
tangents  may  move  further  and  further  out  of  the  field  as  in 
the  parabola.  These  two  kinds  of  infinite  branches  may  be 
called  hyperbolic  and  'parabolic. 

The  character  of  each  of  the  infinite  branches  of  a  curve 
can  always  be  determined  when  the  equation  of  the  curve  is 
known. 

128.  Definition  of  a  rectilinear  asymptote.  If  the  tangents 
at  successive  points  of  a  curve  approach  a  fixed  straight  line 
as  a  limiting  position  when  the  point  of  contact  moves 
further  and  further  along  any  infinite  branch  of  the  given 
curve,  then  the  fixed  line  is  called  an  asymptote  of  the  carve. 

This  definition  may  be  stated  more  briefly  but  less  pre- 
cisely as  follows :  An  asymptote  to  a  curve  is  a  tangent 
whose  point  of  contact  is  at  infinity,  but  which  is  not  itself 
entirely  at  infinity. 

DETERMINATION  OF  ASYMPTOTES 

129.  Method  of  limiting  intercepts.  The  equation  of  the 
tangent  at  any  point  (^j ,  y^)  being 

221 


222 


DIFFER  EN  TIA  L   CA  L  C  UL  US 


[Ch.  XIV 


the  intercepts  made  by  this  line  on  the  coordinate  axes  are 


y^  =  y\~  H 


Xn  —  Xi 


Vi 


dx^ 
dy\ 


(1) 


Suppose  the  curve  has  a  branch  on  which  re  =  oo  and 
y  =  00  ;  then  from  (1)  the  limits  can  be  found  to  which  the 
intercepts  a^^,  y^  approach  as  the  coordinates  a;^,  y^  of  tlie 
point  of  contact  tend  to  become  infinite.  If  these  limits  be 
denoted  by  a,  6,  the  equation  of  the  corresponding  asymptote  is 


a     0 

Ex.  1.    Find    the  ^asymptotes   of 

the 

curve 

y2  =  4  a:2  +  2  a;  +  6. 

Since           dy^^_x^ 
dx           y 

hence  .„,  =  .-y^,=i4±^ 
dy            4  X  +  1 

—  ~  *  ~  ^j  and  thin  —      ^ 

4x  +  1                           4 

when      a;  =  CO. 

du      w2  —  4  a;2  —  x 

dx                y 

x  +  6 

^4  x^  +  2  z  +  6 

To  evaluate  this  expression,  square 
both  terms,  and  then  apply  the  rule  of 
Alt.  73.  The  value  of  the  square  is  \; 
thus,  ^0  =  ±  z- 

Hence  the  asymptotes  are 


Fiu.  aa. 


X     I    y 


-^  +  ^  =  1. 


i.e.,  y  =  2x+l,   y  =  -2x-i. 


129.]  ASYMPTOTES  223 

Ex.  2.    Find  the  equations  of  the  asymptotes  of  the  curve 

Here-  ^^_2x  +  3,/  +  3. 

dx         Sx  +  i  y  —  2 
hence  substituting  in  (1),  and  omitting  the  subscripts  throughout  the 
right-hand  member, 

_2(x^  +  Sxi,  +  2y^)+Sx-2y 
^»~  ■6x  +  iy-2  • 

Replacing  x"^  +  3 xy  +  2 y^  by   -3x  +  2y  +  1  from  the  given  equar 
tion,  this  becomes 

-  3  X  +  2 ;/  +  2 


yo  = 


-»-Kf)-^ 


3a:  +  4^  —  2 


3+4 


\x/      X 


y 

Next,  to  find  the  limit  of  -  as  y  =  oo,  x  =  oo,  observe  that  the  terms 
Sx,2y,  1  are  infinities  of  a  lower  order  (1  is  an  infinite  of  order  0)  than 
x^,  xy,  y"^ ;  hence,  for  large  values  of  x  and  y,  the  terras  of  the  second 
degree  would  have  most  effect  in  fixing  the  form  of  the  curve ;  and  in 
the  limit,  when  x  =  go  and  y  =  vd,  the  smaller  terms  can  be  neglected. 
Then  the  equation  becomes 

x2  +  3x^  +  2y2_o, 

(x  +  2y)(x  +  2^)=0, 


(f-5)(l-0=»- 


Hence,  on  one  branch  ^  = ,  and  on  the  other,  "  =  _  1, 

X         2  X 

Using  these  limiting  values  for  ^  in  the  values  of  y„, 

-3  +  2(-i)^_^"^^^  ^-3  +  2(-l)^ 
^"        3  +  4(-J)  ^"        3  +  4(-l) 

on  the  respective  branches. 

Similarly  for  the  x-intercept,  after  reduction, 
_-3x  +  2y-2 
*"        2X  +  3//  +  3 

=  5,  when  ^  =  —  1 ;  and  =  -  8,  when  ?-  =  —  =• 
X  X        2 


224  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

The  equations  of  the  asymptotes  are  therefore 

i.e,  x  +  y  =  5,       x  +  2y  +  8  =  0. 

Except  in  special  cases  this  method  is  usually  too  compli- 
cated to  be  of  practical  use  in  determining  the  equations  of 
the  asymptotes  of  a  given  curve.  There  are  three  other 
principal  methods,  of  which  at  least  one  will  alwjiys  suffice 
to  determine  the  asymptotes  of  .  curves  whose  equations 
involve  only  algebraic  functions.  These  may  be  called  the 
methods  of  inspection,  of  substitution,  and  of  expansion. 

130.  Method  of  inspection.  Infinite  ordinates,  asymptotes 
parallel  to  axes. 

When  an  algebraic  equation  in  two  coordinates  x  and  y  is 
rationalized,  cleared  of  fractions,  and  arranged  according  to 
powers  of  one  of  the  coordinates,  say  ?/,  it  takes  the  form 

ay"  +  (bx  +  c)3/"-^  +  Cda^  +  ex  +/)^"-2  +  •••  +  w„_i^  +  w„  =  0, 

in  which  w„  is  a  polynomial  of  the  degree  n  in  terras  of  the 
other  coordinate  x. 

When  any  value  is  given  to  x,  the  equation  gives  n  values 
to  1/. 

Let  it  be  required  to  find  for  what  value  of  x  the  corre- 
sponding, ordinate  i/  has  an  infinite  value. 

Suppose  at  first  that  the  term  in  y"  is  present ;  in  other 
words,  that  the  coefficient  a  is  not  zero.  Then  when  any 
finite  value  is  given  to  x,  all  of  the  n  values  of  t/  are  finite, 
and  there  are  thus  no  infinite  ordinates  for  finite  values  of 
the  abscissa. 

Next  suppose  that  a  is  zero,  and  b,  c  not  zero.  In  this 
case  one  value  of  y  is  infinite  for  every  finite  value  of  x,  and 


129-130.]  ASYMPTOTES  225 

thus  one  branch  of  the  curve  lies  entirely  at  infinity.  It  is 
shown  in  projective  geometry  that  this  brancli  always  has 
the  form  of  a  straight  line.  In  this  work  no  account  will  be 
taken  of  such  branches,  and  the  wording  of  the  theorems 
will  in  no  case  refer  to  them. 

There  is  one  particular  value  of  x  that  gives  one  additional 
infinite  value  to  i/,  namely,  the  value  a;  =  —  - ;  for  this  makes 

bx  +  c  (the  coefficient  of  the  highest  power  of  y)  zero,  and 
hence  from  the  theory  of  .equations  one  corresponding  value 
of  1/  must  be  infinite;  and  this  value  is  finite  when  x^  —  -- 

0 

The  equation  of  the  infinite  ordinate  is  bx  -{-  c  =  0. 

Again,  if  not  only  a,  but  also  b  and  c,  are  zero,  there  are 
two  values  of  x  that  make  i/  infinite ;  namely,  those  values 
of  X  that  make  da^  -\-  ex  -^f=  0,  and  the  equations  of  the 
infinite  ordinates  are  found  by  factoring  this  last  equation  ; 
and  so  on. 

Similarly,  by  arranging  the  equation  of  the  curve  accord- 
ing to  powers  of  x,  it  is  easy  to  find  what  values  of  i/  give  an 
infinite  value  to  x. 

Ex.  3.  In  the  curve 

2x^  +  x^y  +  x'tp'  =  a;^  —  y^  —  5, 
find  the  equation  of  the  infinite  ordinate,  and  determine  the  finite  point 
in  which  this  line  meets  the  curve. 

This  is  a  cubic  equation  in  which  the  coefficient  of  y^.  is  zero. 
Arranged  in  powers  of  y  it  is 

y'^{x  +  1)  +  yx2  +  (2 ar8  -  a;2  +  5)  =  0. 

When  a:  =  —  1,  the  equation  for  y  becomes 

0  .  ^2  +  y  ^.  2  =  0, 

the  two  roots  of  which  are  y,  =  co,  y  =  —  2 ;  hence  the  equation  of  the 

infinite  ordinate  is  a:  +  1  =  0.     The  infinite  ordinate  meets  the  curve 

again  in  the  finite  point  (—1,  —  2). 

Since  the  term  in  x^  is  present,  there  are  no  infinite  values  of  x  for 
finite  values  of  y. 


226  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

Ex.  4.   In  the  curve 

x^  +  5xy^  +  2x^  =  3  xhj  +  6, 

find  what  values  of  x  make  y  infinite  and  what  values  of  y  make  x 
infinite. 

131.  Infinite  ordinates  are  asymptotes.  Applying  to  the 
general  equation  of  the  last  article  the  method  of  Art.  129, 

the  slope  of  the  tangent  at  (x,  y)  \^  -M. 

dx 

_     5y"-^  +  (2  (^a; + e)y"-^  +  ■ . . 

~'~wa^"-^+(n-l)(6a;  +  c)y"~^+(w-2)((^r^+gx+/)^«-»+...* 

Now,  the  first  condition  that  y  may  become  infinite  for  a 
finite  value  of  a;,  is  a  =  0  ;  but  when  a  is  zero,  x  finite,  and  y 
infinite,  the  numerator  is  an  infinite  of  higher  order  than 

the  denominator,  hence  -^=oo,  when  x  =  —  -  and  y  =  cc. 

dx  0 

Therefore  the  inclination  of  the  tangent  approaches  nearer 
and  nearer  to  90°,  and  the  tangent  approaches  to  coincidence 
with  the  ordinate  through  the  point  x  =  —-',  and  thus  this 

0 

line  is  an  asymptote  parallel  to  the  ^-axis. 

Similarly,  if  the  value  y  =  k  gives  an  infinite  value  to  x, 
then  the  line  ^  =  A:  is  an  asymptote  parallel  to  the  a:-axis. 

ThuSv  to  determine  all  the  asymptotes  parallel  to  the 
y-axis,  equate  to  zero  the  coeificient  of  the  highest  power  of 
y,\i  it  be  not  a  constant.  If  this  equation  be  of  the  first 
degree,  it  represents  an  asymptote  parallel  to  the  y-axis.  If 
it  be  of  higher  degree,  it  may  be  resolved  into  first  degree 
equations,  each  of  which  represents  such  an  asymptote. 

Similarly,  to  determine  all  the  asymptotes  parallel  to  the 
a;-axis,  equate  to  zero  the  coefficient  of  the  highest  power  of 
a;,  if  it  be  not  a  constant. 


130-1320 


ASYMPTOTES 


227 


Ex.  5.  In  the  curve  a^x  =  y(x  —  d)\  the  line  y  =  0  is  an  asymptote 
coincident  with  the  x-axis,  and  the  line  x  =  a  is  an  asymptote  parallel  to 
the  y-axis. 


Fig.  40. 


Ex.  6.   Find  the  asymptotes  of  the  curve  x^  (^  —  a)  +  xy'^  =  a*. 

132.  Method  of  substitution.     Oblique  asymptotes.     The 

asymptotes  that  are  not  parallel  to  either  axis  can  be  found 
by  the  method  of  substitution,  which  is  applicable  to  all 
algebraic  curves,  and  is  of  especial  value  when  the  equation 
is  given  in  the  implicit  form 

/(;r,y)  =  0.  (1) 

Consider  the  straight  line 

t/=mx  +  b,  (2) 

and  let  it  be  required  to  determine  m  and  b  so  that  this  line 
shall  be  an  asymptote  to  the  curve /(a;,  y)  =  0. 

Since  an  asymptote  is  the  limiting  position  of  a  line  that 
meets  the  curve  in  two  points  that  tend  to  coincide  at  in- 
finity, then,  by  making  (1)  and  (2)  simultaneous,  the  result- 
ing equation  in  x, 

f(x,  mx-hb)  =  0, 

is  to  have  two  of  its  roots  infinite.  This  requires  that  the 
coefficients  of   the  two  highest  powers  of  x  shall  vanish. 


DIFF.  CALC. 


16 


228 


DIFFERENTIAL   CALCULUS 


[Ch.  XIV. 


These  coefficients,  equated  to  zero,  furnish  two  equations, 
from  which  the  required  values  of  m  and  b  can  be  deter- 
mined ;  and  these  values,  substituted  in  (2),  will  give  the 
equation  of  an  asymptote. 

Ex.  7.   Find  the  asymptotes  to  the  curve  y^  =  x^  (2  a  —  x). 

In  the  first  jjlace,  there  are  evidently  no  asymptotes  parallel  to  either 
of  the  coordinate  axes.  To  determine  the  oblique  asymptotes,  make  the 
equation  of  the  curve  simultaneous  with  y  =  mx  +  b,  and  eliminate  y, 
then 

(inx  +  6)8  =  x2(2  a  —  x\ 

or,  arranged  in  powers  of  x, 

(1  +  m3)  x*  +  (3  m%  -  2  a)  a;2  +  3  hhnx  -^¥  =  0. 
Let  r/js  +  1  =  0     and     3m%-2a  =  0, 


then 


hence 


fn  =  -l,     o  =  -o-' 


y  =  -x  + 


2a 


is  the  equation  of  an  asymptote. 

The  third  intersection  of  this  line  with  the  given  cubic  is  found  from 
the  equation  3  rnb^x  +  6*  =  0. 

Y 


132-133.] 
whence 


ASYMPTOTES 
_      2a 


229 


Tliis  is  the  only  oblique  asymptote,  as  the  other  roots  of  the  equation 
for  m  are  imaginary. 

Ex.  8.    Find  the  asymptotes  to  the  curve  y(a^  +  x^)  =  a\a  -  x). 

Y 


Fio.  42. 

Here  the  line  y  =  0  is  a  horizontal  asymptote  by  Art.  130.     To  find 
the  oblique  asymptotes,  put  y  =  mx  ^-h, 
then  (nix  +  6)  (a^  +  x^)  =  a.\a  -  x) ; 

i.e.,  fnx»  +  ^2  +  (ma^  +  a^)  x  +  {a%  -  a*)  =  0, 

hence  m  =  0,     6  =  0,     for  an  asymptote. 

Thus  the  only  asymptote  is  the  line  y  =  0,  already  found. 

133.  Number  of  asymptotes.  The  illustrations  of  the  last 
article  show  that  if  all  the  terms  be  present  in  the  general 
equation  of  an  wth  degree  curve,  then  the  equation  for 
determining  m  is  of  the  nth  degree,  and  there  are  accordingly 
n  values  of  m,  real  or  imaginary.  The  equation  for  finding 
h  is  usually  of  the  first  degree,  but  for  certain  curves,  when 
y  has  been  replaced  by  mx  ■+-  h,  one  or  more  values  of  m,  say 
wij,  may  cause  the  coefiicient  of  a^  and  af""^  both  to  vanish, 
irrespective  of  h.  In  such  cases  any  line  whose  equation  is 
of  the  form  y  =  m^x  +  c  will  satisfy  the  definition  of  an 
asymptote,  independent  of  c ;  but  by  equating  the  coefficient 
of  a^^  to  zero,  two  values  of  h  can  be  found  such  that  the 
resulting  lines  have  three  points  at  infinity  in  common  with 
the  curve.     These  two  lines  are  parallel ;  and  it  will  be  seen 


230  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

that  in  each  case  in  which  this  happens  the  equation  defining 
m  has  a  double  root,  so  that  the  total  number  of  asymptotes 
is  not  increased.  Hence  the  total  number  of  asymptotes, 
real  and  imaginary,  is  in  general  equal  to  the  degree  of  the 
equation  of  the  curve. 

It  is  to  be  observed,  however,  that  in  special  cases  (i.e., 
for  certain  special  values  of  the  given  coefficients)  two  or 
more  of  these  lines  may  coincide,  and  moreover  that  some  of 
these  n  "tangents  at  infinity"  may  be  situated  entirely  at 
infinity  and  thus  be  improperly  called  asymptotes. 

Since  the  imaginary  values  of  m  occur  in  pairs,  it  is  evident 
that  a  curve  of  odd  degree  has  an  odd  number  of  real  asymp- 
totes ;  and  that  a  curve  of  even  degree  has  either  no  real 
asymptotes  or  an  even  number.  Thus,  a  cubic  curve  has 
either  one  real  asymptote  or  three;  a  conic  has  either  two 
real  asymptotes  or  none. 

134.  Method  of  expansion.  Explicit  functions.  Although 
the  two  foregoing  methods  are  in  all  cases  sufficient  to  find 
the  asymptotes  of  algebraic  curves ;  yet  in  certain  special 
cases  the  oblique  asymptotes  are  most  conveniently  found 
by  the  method  of  expansion  in  descending  powers.  It  is 
based  on  the  following  principle  :  a  straight  line  will  be  an 
asymptote  to  a  curve  when  the  difference  between  the  ordi- 
nates  of  the  curve  and  of  the  line,  corresponding  to  a  com- 
mon abscissa,  approaches  zero  as  a  limit  as  the  abscissa 
becomes  larger  and  larger. 

It  will  appear  from  the  process  of  applying  this  principle 
that  a  line  answering  the  condition  just  stated  will  also 
satisfy  the  original  definition  of  an  asymptote. 

Suppose  that  the  equation  of  the  given  curve  can  be  solved 
for  y  in  the  form  of  a  descending  series  of  powers  of  x, 


133-134.]  ASYMPTOTES  231 

beginning  with  the  first  power,  and  let  the  equation 
then  be 

y  =  a,x  +  a,+^  +  ^+..,.  (1) 

The  line  whose  equation  is 

y  =  a^x  +  aj  (2) 

is  an  asymptote  to  the  curve  represented  by  (1) ;  for  the 
difference  between  the  ordinate  of  the  curve  and  line,  corre- 
sponding to  the  same  abscissa  x,  is 

X       ar 

which  approaches  zero  when  a;  =  oo. 

It  is  also  evident  that  the  line  (2)  satisfies  the  original 
definition  of  an  asymptote ;  for,  from  (1),  the  slope  of  the 
tangent  at  the  point  whose  abscissa  is  a;,  is 

and  the  intercept  made  by  the  tangent  on  the  y-axis  is 

dy           ,  2«„  , 
y-x^=a^-\ 2+..., 

ax  X 

hence  when  a;  =  oo,  the  slope  approaches  the  limit  a^,  and 
the  intercept  =  a^ ;  thus  the  equation  of  the  asymptote  is 

y  =  aQX-\-  ay 
Ex.  9.   Find  the  asymptotes  of  the  curve 


1 

The  line  ar  =  1  is  an  asymptote  parallel  to  the  y-axis. 

To  obtain  the  oblique  asymptotes,  write  the  equation  in  the  form 


y 


^'62  t)lfFERENTlAL   CALCULUS  [Ch.  XIV. 

Hence  the  two  oblique  asymptotes  are 

2/  =  ±(x  +  0- 


Fig.  48. 


31 


The  sign  of  the  term  -  -  shows  that  when  x  i  +  oo,  the  curve  is  above 
8  X 
the  first  asymptote,  and  below  the  second,  as  in  figure;  and  that  when 
x  =  —  cc,  the  curve  is  below  the  first  asymptote,  and  above  the  second. 


134.] 


ASYMPTOTES 


233 


The  principal  value  of  the  method  of  expansion  is  that 
it  exhibits  the  manner  in  which  each  infinite  branch  ap- 
proaches its  asymptote. 

Ex.  10.   Find  the  asymptotes  of  the  curve 


Here 


.^(x-i)(2-xy 

X  —  3 


4-D(-ir 


Fig.  44. 


Hence  the  oblique  asymptotes  are 

y=±(x-l). 
The  same  method  may  be  applied  to  cases  in  which  a;  is  an  explicit 
function  of  y. 


234  DIFFERENTIAL   CALCULUS  [Ca  XIV; 

Ex.  11.   Find  the  asymptotes  of 

Here  ^' =  K^ +|)'(^  +  ?)' 

Hence  the  asymptotes  are  x  =  ±  (y  +  2).     The  next  term  —  shows  that 
when  ?/  =  +  CO,  the  curye  is  to  the  right  of  the  first  asymptote,  and  to 


Fio.  45. 


the  left  of  the  second ;  and  vice  versa  when  y  ==  —  co.     The  form  of  ihe 
equation  shows  that  the  curve  has  a  horizontal  asymptote  y  =  0. 

135.   Method  of  expansion.     Implicit  functions.     It   was 

sliown  in  Art.  132  that  the  direction  of  each  oblique  asymp- 
tote is  determined   by  equating  eacli  factor  of  the   terms 


134-135.]  ASYMPTOTES  235 

of  highest  degree,  in  the  equation  of  the  curve,  separately 
to  zero.  The  subsequent  procedure  will  be  shown  by  an 
example. 

Ex.  1.  Determine  the  asymptotes  to  the  curve 

y*-x*-2ax^-b^x  =  0, 

and  the  manner  in  which  the  corresponding  branch  of  the  curve  ap- 
proaches each. 

The  terms  of  highest  degree  are  y*  —  x*,  and  this  expression  has  but 
two  real  linear  factors,  hence  the  curve  cannot  have  more  than  two  real 
asymptotes ;  and  these  are  parallel  to  the  lines  y  ±  a;  =  0.  To  find  the 
asymptote  parallel  to  y  —  x  =  0,  arrange  the  equation  of  the  curve  thus : 

2  ax'^y  +  b'^x 


(1) 


*         (^■'  +  y'Ky  + 

X) 

-14 

"(>4:)(^ 

0 

When  y, 

X  becomes  infinite,  -  =  1 ;  hence 

X 

^    y     b^ 

2  a  ^  +  -, 

Inn                 X      x^ 

2a 

a 

^""o-aif-o' 

4  " 

-^ 

and  the  equation  of  the  asymptote  is 

y  =  a:  +  r- 

(2) 


(3) 


To  obtain  the  next  term  in  the  equation  of  the  curve,  use  (3)  as  a 
first  approximation,  which  gives 

2^  =  1  +  — ,  correct  as  far  as  the  order  -> 
X  2x  X 

^=(l  +  — y=  1  +  -,  to  the  same  order;        (4) 
x^      \        2x/  X 

1  +  2  =  2  +  ^  =  2(1+-^). 
X  2x         \        ixj 

l  +  g  =  2  +  «  =  2(l  +  f). 
a^  X         \        2xJ 


236  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

These  values  substituted  in  (1)  give  as  a  second  approximation 

Hence  the  curve  approaches  the  lower  side  of  the  asymptote  on  the 
right,  and  the  upper  side  on  the  left. 

Similarly   the    equation    of    the    branch    approaching   the   direction 
^  +  z  =  0  will  be  found  to  have  the  successive  approximations 
,   a  ,  a   ,   1  a* 

^  2*^  28a; 

and  thus  on  the  right  the  curve  approaches  the  upper  side  of  the  asymp- 
tote, and  on  the  left,  the  lower  side. 

If  the  term  in  -  should  happen  to  disappear  from  the  result,  a  third 

^  .  .  1    . 

approximation  may  be  obtained  by  keeping  the  terms  of  order  —  in  the 

equatioiis  that  correspond  to  (1),  (4),  (5),  (6). 
Ex.  2.    2/3  -  x2^  +  2  2/2  +  4  y  +  a;  =  0. 
Ex.  3.   x8  -t-  2  a;2?/  -  a:3/2  -  2  2/3  +  4  3/2  +  2  xy  +  y  =  1. 
Ex.  4.   y^  =  x^  +  (fix. 

136.  Curvilinear  asymptotes.  When  two  curves  are  so 
situated  that  the  difference  between  their  ordinates  corre- 
sponding to  the  same  abscissa  approaches  zero  as  a  limit 
when  the  common  abscissa  is  made  larger  and  larger,  then 
each  curve  is  said  to  be  an  asymptote  of  the  other.  This 
definition  will  also  apply  if  the  words  "ordinate"  and 
"  abscissa  "  be  interchanged. 

E.g..,  suppose  that  the  equation  of  a  given  curve  can  be 
brought  to  the  form 

X        <J^        7? 

then  it  follows  from  the  definition  that  the  curve 


135-137.3 


ASYMPTOTES 


237 


is  a  second  degree  asymptote  to  the  given  curve  ;  and 

y  =  ax^  +  hx  -\-  c  -\- -  ^ 

X 

i.e.,  xy  =  a^  +  h:i^  -\-  ex  -\-  d 

is  a  third  degree  asymptote,  and  so  on. 

Ex.   Find  the  second  and  third  degree  asymptotes  to  the  curves  of 
examples  8-11,  Arts.  132-134. 

137.  Examples  of  asymptotes  of  transcendental  curves. 
1.    Consider  the  curve 

y^Xogx. 
Here,  when 

a;=0, 
«/  =  —  00, 


and 


dx 


Fie.  48. 


hence  the  line  re  =  0  is  an  asymp- 
tote, by  Art.  131. 

2.    The    exponential   curve    y  =  e^.     In   this   case,  when 

a;  =  —  Qo,  y  =0,  -^  =  0.     Hence  ^  =  0  is  an  asymptote. 


Fig.  47. 


238 


DIFFERENTIAL  CALCULUS 


[Ch.  XIV. 


3.    Find  the  asymptotes  to  the  curve  1  +  t/  =  e^. 

When  X  approaches  zero  from  the  positive  side,  ?/  =  +  cx), 

and  -^  =  4-  00  ;  but  when  x  approaches  zero  from  the  nega- 
dx  , 

tive  side,  a;  =  0,  and  — ^  =  0.      Hence  the  line  x  =  0  is  an 
ax 


Fig.  48. 


asymptote  at  y  =  +  qo  on  the  positive  side  of  the  ?/-axis. 
Again,  when  a;  =  ±ao,  y  =  Q  ;  hence  the  line  y  =  0  is  an 
asymptote  both  at  a;  =  +  oo  and  —  oo. 


4.    The  probabil- 


ity curve, 


6.    The  curve 


2      2^-1 

y  =—^- 


Pig.  50. 


137-138,] 


ASYMPTOTES 


239 


EXERCISES 
Find  the  asymptotes  of  the  following  curves : 

1.  (x  +  a)f  =(!/  +  h)x^.  8.    (x  -  2  a)y^  =  x^  -  cfi. 

2.  xY+ax(x  +  yy-2aY—a*  =  0.      9.  y'^  =  xX2a-x). 

3.  X*y*  -  (X2  -  2,2)2  ...  3,2  _  1  ^  0.         10.     y(a2  +  X2)  ::=  0^(0  -  x). 

4.  (x2-y2)2_4y2  +  y^0.  11      X^2  +  ^^2  =  „8. 


5.  x\x  -y)--  a2(a;2  +  y^)  =  0. 

6.  y 


x2  +  3a2 


la 


X8 


12.  (X2  +  «2)j.2  ^  (-a2  _  3.2)^2 

13.  X2^2  ^  ^^  .,.  3.  +  y, 

14.  X2^2  =  (a  +  y)2(^ft2  _  ^2). 

15.  ^(x  -  rjY  =  y(x  -  ij)+2. 


138.   Asymptotes   in    polar   coordinates.      When   a   curve 
defined  by  an  equation  in  polar  coordinates  has  an  asymptote, 
this  line  must  be  paral- 
lel to  the  radius  vector 
to  the  point  at  infinity 
on  the  curve. 

In  Fig.  51,  consider 
the  curve  KP'P,  hav- 
ing the  asymptote  PT. 
The  radius  vector  to 
the  point  at  infinity 
must  be  parallel  to  the 
asymptote,  for  these 
two  lines  must  inter- 
sect at  infinity  ;  and,  moreover,  the  asymptote,  according  to 
the  definition  in  Art.  128,  must  pass  within  a  finite  distance 
of  this  radius  vector. 

The  polar  subtangent  OM,  being  by  definition  perpendicu- 
lar to  the  radius  vector  OP,  Avill,  when  P  passes  to  infinity, 
become  a  common  perpendicular  to  the   radius  vector   OP 


Fig.  51. 


240  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

and  to  the  asymptote  MP  ,  hence  the  measure  of  the  com- 
mon perpendicular  is 

lim   f  ^dd\ 

139.  Determination  of  asymptotes  to  polar  curves.  To  de- 
termine whether  a  given  curve  has  asymptotes,  first  find  for 
what  values  of  0,  the  vector  p  becomes  infinite  ;  then  substi- 
tute each  of  these  values  of  6  in  the  expression  for  the  polar 
subtangent.  If  the  result  of  any  such  substitution  is  finite, 
there  is  a  corresponding  asymptote. 

To  construct  the  asymptote,  look  along  the  direction  of 
the  infinite  radius  vector  from  the  pole,  and  turn  through 

T  /Iff 

a  right  angle,  to  the  right  if  V"^  p^—-  he  positive,  and  to  the 
left  if  it  be  negative  (Art.  120).  Measure  a  distance  from 
the  pole  in  this  perpendicular  direction  equal  to    V°    p^ — , 

and  through  its  extremity  draw  a  line  parallel  to  the  infinite 
radius  vector ;  this  line  will  be  the  required  asymptote. 

Circular  asymptotes.  In  some  cases  it  may  happen  that  when 
B  is  made  larger  and  larger  without  limit,  the  value  of  p  may 
approach  a  definite  limit  a;  thus  q^^^P  =  ("'•  The  circle 
whose  equation  is  p  =  a  is  then  called  an  asymptotic  circle. 

E.g.     The    curve    p  =  .      ^^"  .  has  an  asymptotic  circle  p  =  1:  the 
6  +  cos  d 
curve  being  exterior  to  the  circle  from  the  middle  of  the  first  quarter  to 
the  middle  of  the  third  quarter,  and  interior  for  the  remainder  of  the 
circle ;  it  approaches  nearer  to  the  circle  with  every  revolution  of  6. 

Ex.  1.   Find  the  rectilinear  asymptotes  to  the  curve  p  =  -: — -. 

When  6  =  0,  p  =  a ;  but  when  6  =  mr,n  being  any  positive  or  nega- 
tive integer,  p  becomes  infinite. 

Since  dp  ^a(sm6- 6  cos  ef)^ 

dd  sin^d 

hence  P^^-— Z-^^-^- 

dp     sin  9-6  cos  Q 


138-139.] 


ASYMPTOTES 


241 


When  6  =  rnr,  this  expression  becomes  amr  or  —  antr,  according  as  n 
is  odd  or  even,  and  may  therefore  be  written  in  the  form  (—  l)"~ia7j'!r. 

There  are  thus  an  infinite  number  of  asymptotes,  all  parallel  to  the 
initial  line,  and  situated  at  intervals  air  from  each  other. 

When  n  is  positive,  the  asymptotes  are  above  the  initial  line ;  when  n 
is  negative,  they  are  below  it.     There  are  no  circular  asymptotes. 


In  many  problems  it  shortens  the  work  to  substitute  - 

for  p  in  the  equation  of  the  curve,  and  then  to  find  what 

d0 
values  of  0  will  make  u  vanish.     The  expression  p^—-  for 

dp 

the  length  of  the  polar  subtangent  then  becomes  —  — ;  and 

hence  „  ^q( ),  taken  for  any  of  the  values  of  6  just  found, 

measures  the  distance  of  the  corresponding  asymptote  from 
the  pole. 


Ex.  2.   Find  the  asymptotes  to  the  curve 

p sin  46  =  a  sin  3 $. 


Put  p  =  - ,  then 
u 


sin  4  ^  =  au  sin  3  6, 


242 


DIFFERENTIAL   CALCULUS  [Cn.  XIV.  139. 


and  «=:0,  when      ^  =  ±7,     ±~,     ±'^, 
4  2  4 


-f 


By  differentiation,  4  cos  4  ^  =  «  —  sin  3  ^  +  3  «m  cos  3  $, 


du 
dd 

This  expression  becomes 
corresponding  asymptote  is 


4  cos  4^  +  3  au  cos  3  6 


a  sin  dd 


4\/2 


4V2 


Fig.  53. 


wlien  ^  =  — ;  hence  the  distance  to  the 
4 

To  construct  the  asyinptote,  look 
from  the  pole  along  the  direction  of 

45°,  measure  a  distance units  to 

4V2 
the  right,  perpendicular  to  this  radius 
vector ;  then  draw  a  line  through  the 
end  of  the  perpendicular,  parallel  to 
the  infinite  radius  vector  (Fig  53). 
The  student  should  determine  the 
number  and  position  of  the  remaining 
asymptotes. 

EXERCISES 


Find  and  draw  the  asymptotes  to  the  following  curves 

1.  The  reciprocal  spiral  p6  =  a. 

2.  p  cos  6  =  a  cos  2  6.  4 

3.  p  =  b  sec  aO. 
6.   Show  that  the  curve  p  = 


p  cos  29  =  a  sin  3  6. 
5.   p(e^  -  1)  =«(e»  +  1). 
has  no  asymptote. 


1  —  cos  6 

7.  Show  that  the  initial  line  is  an  asymptote  to  two  branches  of  the 
curve  p2  sin  $  =  a^  cos  2  6. 

8.  Find  the  rectilinear  and  circular  asymptotes  of  the  curve 


''      6^-1 
9.    Which  of  the  curves  in  1-7  have  circular  asymptotes? 


CHAPTER   XV 


DIRECTION   OF   BENDING.      POINTS   OF   INFLEXION 


140.  Concavity  upward  and  downward.  A  curve  is  said 
to  be  concave  downward  in  the'  vicinity  of  a  point  P  when, 
for  a  finite  distance  on  each  side  of  P,  the  curve  is  situated 


Fig.  54. 

below  the  tangent  drawn  at  that  point,  as  in  the  arcs  AD, 
I'll.  It  is  concave  upward  when  the  curve  lies  above  the 
tangent,  as  in  the  arcs  DF,  HK. 

It  is  evident,  by  drawing  successive  tangents  to  the  curve, 
as  in  the  figure,  that  if  the  point  of  contact  advances  to  the 
right,  the  tangent  swings  in  the  positive  direction  of  rotation 
when  the  concavity  is  upward,  and  in  the  negative  direction 
when  the  concavity  is  downward.  Hence  upward  concavity 
may  be  called  a  positive  bending  of  the  curve,  and  down- 
ward concavity,  negative  bending. 

A  point  at  which  the  direction  of  bending  changes  con- 
tinuously from  positiv^e  to  negative,  as  at  F,  is  called  a  point 

DIFF.  CALC.  17  243 


"ZU  DIFFERENTIAL   CALCULUS  [Ch.  XV. 

of  inflexion,  and  the  tangent  at  such  a  point  is  called  a 
stationary/  tangent. 

The  points  of  the  curve  that  are  situated  just  before  and 
just  after  the  point  of  inflexion  are  thus  on  opposite  sides  of 
the  stationary  tangent,  and  hence  the  tangent  crosses  the 
curve,  as  at  D,  F,  H. 

141.   Algebraic  test  for  positive  and  negative  bending.     Let 

the  inclination  of  the  tangent  line,  measured  from  the  right- 
hand  end  of  the  »;-axis  toward  the  forward  (right-hand)  end 
of  the  tangent,  be  denoted  as  usual  by  0,  then  <^  is  an  in- 
creasing or  decreasing  function  of  the  abscissa  according  as 
the  bending  is  positive  or  negative ;  for  instance,  in  the  arc 

AD,  the  angle  </>  diminishes  from  -\-  —  through  zero  to ; 

Z  4 

in    the   arc   DF^   <^   increases  from    — -    through  zero  to 

o" ;    in  the  arc  FH,  <f)   decreases   from    +  —    through    zero 

to  —  — ;  and  in  the  arc  ffK,  ^  increases  from  —  —  through 

zero  to  -f— • 
4 

At  a  point  of  inflexion  <^  has  evidently  a  turning  value 
which  is  a  maximum  or  minimum,  according  as  the  concavity 
changes  from  upward  to  downward,  or  conversely. 

Thus  in  Fig.  54,  ^  is  a  maximum  at  F,  and  a  minimum  at 
D  and  at  IT. 

Instead  of  recording  the  variation  of  the  inclination  (f), 
it  is  generally  convenient  to  consider  the  variation  of  the 
slope  tan  (fi,  which  is  easily  expressed  as  a  function  of  x  by 
the  equation 

Since  tan  <f>  is  always  an  increasing  function  of  (f>,  it  follows 
that,  according  as  the  concavity  is  upward  or  downward,  the 


140-141.]  DIRECTION  OF  BENDING  245 

slope  function  -f-  is  an  increasing  or  a  decreasing  function 
ax 

ot  ar,  and  hence  that  its  a;-derivative  is  positive  or  negative. 

Thus  the  bending  of  the  curve  is  in  the  positive  or  nega- 

cPv 
tive  direction  of  rotation,  according  as  the  function  —4  is 

positive  or  negative. 

At  a  point  of   inflexion  the  slope  -r^  is  a  maximum  or 

minimum ;  and  its  derivative  — ^  changes  sign  from  positive 

dor 

to  negative  or  from  negative  to  positive.  This  latter  con- 
dition is  evidently  both  necessary  and  sufficient  in  order  that 
the  point  (a;,  i/}  may  be  a  point  of  inflexion  on  the  given 
curve c 

Hence,  the  coordinates  of  the  points  of  inflexion  on  the 
curve 

may  be  found  by  solving  the  equations 

and  then  testing  whether  fix)  changes  its  sign  as  x  passes 
through  the  critical  values  thus  obtained.  To  any  critical 
value  a  that  satisfies  the  test,  corresponds  the  point  of 
inflexion  («,/(«)). 

Ex.  1.   For  the  curve 

find  the  points  of  inflexion,  and  show  the  mode  of  variation  of  the  slope 

and  of  the  ordinate. 

Here  ^  =  4  x  (a:^  -  1), 

dx 

hence  the  critical  values  for  inflexions  are  z  = —  —  .58  approxi- 

V3 


246 


DIFFERENTIAL   CALCULUS 


[Ch.  XV. 


niately ;  and  x  —  +  .58.  It  will  be  seen  that  as  x  increases  through  —  ..58, 
the  second  derivative  changes  sign  from  positive  to  negative,  hence  there 
is  an  inflexion  at  which  the  concavity  changes  from  upward  to  down- 
ward. Similarly,  at  a;  =  +  .58  the  concavity  changes  fi-oin  downward  to 
upward.  The  following  numerical  table  will  help  to  show  the  mode  of 
variation  of  the  ordinate  and  of  the  slope,  and  the  direction  of  bending. 

As  X  increases  from  —  co  to  —  .58,  the 
bending  is  positive,  and  the  slope  continually 
increases  from  —  go  through  zero  to  a  maxi- 
mum value,  1.5,  which  is  the  slope  of  the 
stationary  tangent  drawn  at  the  point 
(-.58,  .44). 

As  X  continues  to  increase  from  -  .58  to 
+.58,  the  bending  is  negative,  and  the  slope 
decreases  from  +1.5  through  zero  to  a  mini- 
mum value,  —  1.5,  which  is  the  slope  of 
the  stationary  tangent  drawn  at  the  point 
(+  .58,  .44). 

Finally,  as  x  increases  from  +  .58  to  +  oo ,  the  bending  is  positive, 
and  the    slope    increases    from    the 
value  —  1.5  through  zero  to  +  co. 

The  values  x  =  —  1,  0,  +1,  at 
which  the  slope  passes  through  zero, 
correspond  to  turning  values  of  the 
ordinate. 


X 

y 

(Ix 

dx2 

—  00 

+  GO 

—  00 

+ 

-2 

+  25 

-24 

+ 

-1 

0 

0 

+ 

-.58 

+  .44 

+  1.5 

0 

0 

1 

0 

— 

+  .58 

+  .44 

-1.5 

0 

1 

0 

0 

+ 

+  00 

+  00 

+  00 

+ 

Ex.  2.     Examine    for    inflexions 
the  curve 


V 


J 


Fig.  55. 

In  this  case 

y  =  2  +  (x  +  4)i, 

^  =  l(x  +  4)-*, 
(IX      .3 


Fig.  56. 


dx'         9*^  ^ 

Hence,  at  the  point  (—4,  2), 


dji 

(ix 

and  '^-^  are  infinite.    When  x<  — 4, 
(Ix^ 


—\  is  positive,  and  when  x>  — 4,  ^  is  negative. 
dx^  dx^ 


141-142.] 


DIRECTION  OF  BENDING 


247 


Thus  there  is  a  point  of  inflexion  at  (—4,  2),  at  which  the  slope  is 
infinite,  and  tiie  bending  changes  from  the  positive  to  the  negative 
direction. 

Ex.  3.   Consider  the  curve  y  =  x*. 

'ly.  =  i  x\   '^  =12  xK 
dx  '   dx^ 

At  (0,  0),  — ^  is  zero,  but  the  curve 

dx^  72 

has  no  inflexion,  for  '—^  never  changes 
dx^ 


sign  (Fig.  57). 


Fig.  6T. 


142  Analytical  proof  of  the  test  for  the  direction  of  bend- 
ing. Let  the  equation  of  a  curve  be  ^  =  /(a:),  and  let 
P,  (x^,  yj),  be  a  point  upon  it ;  then  the  equation  of  the 
tangent  at  P  is 

When  X  changes  from  x^  to  x^  +  A,  let  the  ordinate  of  the 

tangent  change  from  y^  to  y', 
and  that  of  the  curve  from  i/^ 
to  ^''  ;  then  it  is  proposed  to 
determine  the  sign  of  the  dif- 
ference of  ordinates  t/"  —  i/'  cor- 
responding to  the  same  abscissa 
x^  +  h. 


(x-h)   X,  (x,+7i) 
Fio.  58. 


By  Taylor's  theorem. 


h^ 


y"  =fix,  +  h)  =f(x{)  +  hf(xO  +  j^/"(^i  +  ^^); 
and  from  the  above  equation  of  the  tangent, 

hence  y'  =  ^i  +  ¥'C^D  =/(^i)  +  ¥'(.^i)y 

and  it  follows  that 


248 


DIFFERENTIAL   CALCULUS 


[Ch.  XV. 


As  h  is  made  smaller  and  smaller, /"(rrj  +  OK)  will  have 
the  same  sign  as/"(a:j);  but  the  factor  A^  is  always  positive, 
hence  when/"(a:;j)  is  positive,  y"  —  y'  is  positive,  and  thus 
the  curve  is  above  the  tangent,  at  both  sides  of  the  point  of 
contact,  that  is,  the  concavity  is  upward.  Similarly  when 
f"(p\)  is  negative,  the  concavity  is  downward. 

This  agrees  with  the  former  result. 

143.  Concavity  and  convexity  towards  the  axis.  A  curve 
is  said  to  be  convex  or  concave  toward  a  line,  in  the  vicinity 
of  a  given  point  on  the  curve,  according  as  the  tangent  at 
the  point  does  or  does  not  lie  between  the  curve  and  the 
line,  for  a  finite  distance  on  each  side  of  the  point  of  contact. 


Fio.  59  a. 


First,  let  the  curve  be  convex  toward  the  a;-axis,  as  in  the 
left-hand  figure  ;  then  if  y  is  positive,  the  bending  is  positive 

72 

and  —^  is  positive  ;  but  if  y  is  negative,  the  bending  is  neg- 

ative  and  -^  is  negative.     Thus  in  either  case  the  product 

y—^  is  positive. 

Next,  let  the  curve  be  concave  toward  the  2:-axis,  as  in 
the  right-hand  figure  ;  then  if  y  is  positive,  the  bending  is 

72 

negative  and  — ^  is  negative  ;  but  if  y  is  negative,  the  bend- 
ed 


142-144.]  DIRECTION   OF  BENDING  249 

ing  is  positive  and  -—^  is  positive.     Thus  in  either  case  the 

TO  CvUy 

product  y-T-^  is  negative.     Hence  : 

In  the  vicinity  of  a  given  point  (x,  y')  the  curve  is  convex  or 
concave  to  the  x-aads,  according  as  the  product  y  — ^  is  positive 

CtiOu 

or  negative. 

EXERCISES 

1.  Show  that  the  curve  y  =  - — has  a  point  of  inflexion  at  the 

origin,  and  also  when  x  =  ±  a  V3. 

2.  In  the  curve  y  («*  —  6*)  =  x  (a:  —  a)*  —  xh*,  there  is  a  point  of 
inflexion  at  x  =  - —     Examine  the  points  at  which  x  =  a. 

3.  Find  the  points  of  inflexion  of  the  curve 

4.  Show  that  the  curve  y  (x^  +  a^)  =  a^  (a  —  x)  has  three  points  of 
inflexion  on  the  same  straight  line. 

5.  Find  the  points  of  inflexion  on  the  curs^e  y^  (x  —  1)  =  x\ 

6.  Show  that  the  curve  6x(l  —  x)y  =  1  +  3x  has  one  point  of 
inflexion,  and  three  asymptotes. 

7.  Show  why  a  conic  section  cannot  have  a  point  of  inflexion. 

8.  Draw  the  part  of  the  curve   ahf  = ax^  +  2  a'  near  its  point  of 

inflexion. 

144.  Concavity  and  convexity ;  polar  coordinates.  A  curve 
referred  to  polar  coordinates  is  said  to  be  concave  or  convex 
to  the  pole,  at  a  given  point  on  the  curve,  according  as  the 
curve  in  the  neighborhood  of  that  point  does  or  does  not  lie 
between  the  tangent  and  the  pole. 

Let  p  be  the  perpendicular  from  the  pole  to  the  tangent 
at  the  point  (/j,  ^).  Then  when  the  curve  is  concave  to  the 
pole,  p  evidently  increases  with  p,  as  in  the  arc  AB,  and 

diminishes  with  p,  as  in  the  arc  BQ  (Fig.  60  a);  hence  ~ 
is  positive  (Art.  20). 


250 


DIFFERENTIAL   CALCULUS 


[Ch.  XV. 


Again,  when  the  curve  is  convex  to  the  pole,  j9  increases 
when  p  diminishes,  as  in  the  arc  DE  (Fig.   60  J),  and  p 

diminishes  when  p  increases,  as  in  the  arc  EF ;   hence  -/- 

dp 
IS  negative. 


Fig.  60  a. 


Fig.  60  Ik 


Thus  the  curve  is  concave  or  convex  to  the  pole  at  the 

point  (/3,  ^),  according  as  -^  is  positive  or  negative. 

To  express  this  condition  in  terms  of  ^-derivatives  of  /o, 
use  the  equation  P  —  P  sin  i/r, 


P 


-^  = -2  CSC2  i/r  = -^  (1  +  C0t2  a/r )  = -2 


dd 


)4® 


1    ''' 


because  tan  -v|r  =  /o-z-,  by  Art.  118. 
dp 

This  may  be  simplified  by  putting  -  =  w,  /a  =  -,  whence 

\        rlti  I 


dp 
dB 


du 


,  and  equation  (1)  becomes 


i^^-^HIJ- 


1 

p' 


(2) 


Differentiation  as  to  u  gives 


du    ^u    dO 


.l.^  =  2w  +  2^ , 

j9^     du  dd     d&^     du 


dp  _ 


du 


=  —  p'^iu  + 


^u\ 

de^/ 


(3) 


144.]  DIRECTION    OF  BENDING  251 

,  dp  _dp     du  _      dp      1  _      dp       2 

dp      du     dp  du     p^         du        * 

hence,  from  (3),         -j-  =  p^u^  i  u  +  j^  )• 

Since  p  is  always  taken  positively,  hence 

The  curve  is  concave  or  convex  to  the  pole  at  the  point  (p,  ^), 

according  as  u  +  — —  is  positive  or  negative. 
d(P 


EXERCISES 
Trace  the  following  curves  near  their  points  of  inflexion  : 
1.  p  =  ^-^    (find  its  asymptotes).      2.   p  =  -^^.       3.  p  =  6^. 

4.   In  the  curve  defined  by  the  two  equations 

X  =  a  (1  —  cos  <^),  y  =  n  (n<f>  +  sin  <^), 

show  that  there  is  an  inflexion  at  the  point  where  cos  d>  = 

n 

5.  Locate  the  inflexions  on  the  curve  p  =  -^^-     (See  Fig.  52.) 

sin  6 

6.  Find  the  coordinates  of  the  inflexion  in  Fig.  40. 

7.  In  Fig.  41,  show  that  the  inflexional  tangent  is  vertical. 

8.  Show  that  there  are  three  real  inflexions  in  Fig.  42. 

9.  How  many  inflexions  are  there  in  Figs.  44,  45? 

10.  In  the  logarithmic  curve,  the  curvature  is  always  negative;  and 
in  the  exponential  curve  it  is  always  positive.     (Figs.  46,  47.) 

11.  Locate  the  points  of  inflexion  in  Figs.  48,  49,  50. 


CHAPTER   XVI 

CONTACT  AND  CURVATURE 

145.  Order  of  contact.  The  points  of  intersection  of  the 
two  curves 

are  found  by  making  the  two  equations  simultaneous  ;  that 
is,  by  finding  those  values  of  x  for  which 

Suppose  x—a\^  one  value  that  satisfies  this  equation,  then 
the  point  x=  a,  y  =  <^(a)  =  ^((t)  is  common  to  the  curves. 

If,  moreover,  the  two  curves  have  the  same  tangent  at  this 
point,  they  are  said  to  touch  each  other,  or  to  have  contact 
of  the  first  order  with  each  other.      The  values  of  y  and 

of  -^  are  thus  the  same  for  both  curves  at  the  point  in 
ax 

question,  and  this  requires  that 

<^  («)  =  -^  (a), 
<^'(a)=>/r'(«). 

TO 

If,  in  addition,  the  values  of  —^  be  the  same  for  each 
curve  at  the  point,  then 

and  the  curves  are  said  to  have  a  contact  of  the  second 
order  with  each  other  at  the  point  for  which  x  =  a. 

If  (f>  (a)  =  yfr  (a),  and  all  the  derivatives  up  to  the  wth 
order  be  equal  to  each  other,  the  curves  are  said  to  have 
contact  of  the  wth  order.     This  is  seen  to  require  n-\-l  con- 

262 


Ch.  XVI.  145-14G.]      CONTACT  AND   CURVATURE  253 

ditions ;  hence  if  the  equation  of  the  curve  y  =  ^(x)  be 
given,  and  if  the  equation  of  a  second  curve  be  written  in 
the  form  y  =  '\lr(x^,  in  which  yjr(x)  proceeds  in  powers  of  x 
with  undetermined  coefficients,  then  w  +  1  of  these  coeffi- 
cients coukl  be  determined  by  requiring  the  second  curve  to 
have  contact  of  the  wth  order  with  the  given  curve  at  a 
given  point. 

146.  Number  of  conditions  implied  by  contact.  A  straight 
line  has  two  arbitrary  constants,  which  can  be  determined  by 
two  conditions  ;  thus,  a  straight  line  can  be  drawn  which 
touches  a  given  curve  at  any  specified  point. 

In  general  no  line  can  be  drawn  having  contact  of  an 
order  higher  than  the  first  with  a  given  curve  ;  but  there 
are  certain  points  at  which  this  can  be  done.  For  instance, 
if  the  equation  of  a  line  be  written  y  =  mx  -(-  5,  then 

ax  ax^ 

hence,  through  any  arbitrary  point  x  =  a  on  a  given  curve 

y=^(x),  a  line  can  be  drawn  which  has  contact  of  the  first 

order  with  the  curve,  but  which  has  not  in  general  contact 

of  the  second  order ;  for  the  two  conditions  for  first  order 

contact  are 

ma  -{-b=  ^(a\ 

m         =  0'(«), 

which  are  just  sufficient  to  determine  m  and  h;  and  the 
additional  condition  for  second-order  contact  is  0  =  (^"(a), 
which  is  satisfied  whenever  the  point  x  =  a  is  a  point  of 
inflexion  on  the  given  curve  y  =  <l>(x).  Thus  the  tangent 
at  a  point  of  inflexion  on  a  curve  has  contact  of  the  second 
order  with  the  curve. 


25'4  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

The  equation  of  a  circle  has  three  independent  constants. 
It  is  therefore  possible  to  determine  a  circle  having  contact 
of  the  second  order  with  a  given  curve  at  any  assigned 
point. 

The  equation  of  a  parabola  has  four  constants,  hence  a 
parabola  can  be  found  which  has  contact  of  the  third  order 
with  the  given  curve  at  any  point. 

The  general  equation  of  a  central  conic  has  five  inde- 
pendent constants,  hence  a  conic  can  be  found  which  has 
contact  of  the  fourth  order  with  a  given  curve  at  any  given 
point. 

As  in  the  case  of  the  tangent  line,  special  points  may  be 
found  for  which  these  curves  have  contact  of  higher  order. 

147.   Contact  of  odd  and  of  even  order. 

Theorem.  At  a  point  where  two  curves  have  contact  of 
an  odd  order  they  do  not  cross  each  other ;  but  they  do 
cross  where  they  have  contact  of  an  even  order. 

For,  let  the  curves  t/  =  <f>  (a:),  y  =  yfr  (^x)  have  contact  of 
the  nth  order  at  the  point  whose  abscissa  is  a ;  and  let  i/^, 
y^  be  the  ordinates  of  these  curves  at  the  point  whose 
abscissa  is  a  +  A ;  then 

y^^i^{a-\-  A),     i/^  =  ylr(a-h  A), 
and  by  Taylor's  theorem 

711  (w+1)! 

y,  =  f(_a}  +  t'(<.)  .  h  +  *^  .  h'  +  - 


146-149.]  CONTACT  AND  CURVATURE  255 

Since  by  hypothesis  the  two  curves  have  contact  of  the 
nth  order  at  the  point  whose  abscissa  is  a, 

hence     <^(«)=-f(a),  <^'(a)= -f' (a),  ...,  <^"{a^=  y\r"(a), 

and        y,-y^  =  j^^,W^Ka  +  eh)-r^\a  +  e,K)-]', 
(n-[-i). 

but  this  expression,  when  h  is  sufficiently  diminished,  has 
the  same  sign  as 

A»  + 1  [</>«  + 1  (a) --,/r" +  !(«)]; 

hence,  if  n  be  odd,  y^  —  y^  does  not  change  sign  when  7i  is 
changed  into  —  A,  and  thus  the  two  curves  do  not  cross  each 
other  at  the  common  point.  On  the  other  hand,  if  n  be 
even,  y^  —  y.^  clianges  sign  with  h ;  and  therefore  when  the 
contact  is  of  even  order  the  curves  cross  each  other  at 
their  common  point. 

For  example,  the  tangent  line  usually  lies  entirely  on  one 
side  of  the  curve,  but  at  a  point  of  inflexion  the  tangent 
crosses  the  curve. 

Again,  the  circle  of  second-order  contact  crosses  the 
curve  except  at  the  special  points,  noted  later,  in  which  the 
circle  has  contact  of  the  third  order. 

148.  Circle  of  curvature.  The  circle  that  has  contact  of 
the  closest  (i.e.^  second)  order  with  a  given  curve  at  a  speci- 
fied point  is  called  the  osculating  circle  or  circle  of  curvature 
of  the  curve  at  the  given  point.  The  radius  of  this  circle  is 
called  the  radius  of  curvature,  and  its  center  is  called  the 
center  of  curvature  at  the  assigned  point. 

149.  Length  of  radius  of  curvature ;  coordinates  of  center 
of  curvature. 

Let  the  equation  of  a  circle  be 


256 


DIFFERENTIAL   CALCULUS 


[Ch.  XVL 


in  which  R  is  the  radius,  and  a,  /8  are  the  coordinates  of  the 
center,  the  current  coordinates  being  denoted  by  JT,  y, 
to  distinguish  them  from  the  coordinates  of  a  point  on  the 
given  curve. 

It  is  required  to  determine  R^  a,  /3,  such  that  this  circle 
may  have  contact  of  the  second  order  with  the  given  curve 
at  the  point  (x,  y). 

From  (1),  by  successive  differentiation, 


\dX)      ^         ^^JX2 


(2) 


If  the  circle  (1)  has  contact  of  the  second  order  at  the 
point  (a;,  if)  with  the  given  curve,  then  the  common  abscissa 
x=  X.  makes 

dX     dx      dX^      dx^\ 


hence,  from  (2),    {x—  «)  +  (?/  —  /3)  -^  =  0, 

dx 


i+(IT+(^-«^=»- 


dx 


whence 


y-^  =  ~ 


1     (dy 

\dx. 
d^y 


X  —  a  = 


dy 
dx 


)<m 


d?y 
d^ 


(4) 


(-^) 


and  finally,  by  substitution  in  (1), 

\dxj  ) 


R  = 


1+m- 


(^>) 


149-150.]  CONTACT  AND  CURTATURB  257 

If,  for  shortness,  w,  n  be  written  for  -^,    — ^,  then  the 

ax      dor 

coordinates  of  the  center  and  the  radius  of  the  circle  of 
curvature  are  given  by  the  equations 

mCl+m^')  a  1  +  m2        ^      (1  +  7»2)5 

X—  a  =  — ^i — ^ ;      V  —  p  = 1 ;      M  =  ^ — ' ^. 

n  n  '         n 

150.  Second  method.  The  osculating  circle  is  sometimes 
defined  as  the  limiting  position  of  a  circle  passing  through 
three  points  on  the  curve  when  two  of  these  points  move 
towards  the  third  as  a  limit. 

It  is  proposed  to  find  the  equation  of  this  circle,  and  thus 
to  show  that  the  two  definitions  lead  to  the  same  result. 

Let  X  —  h,  Xy  X  +  h  be  the  abscissas  of  three  points  on  the 
curve,  and  y  —  k.  y.  y  -\-  y  the  corresponding  ordinates,  in 
which  k'  is  not  in  general  equal  to  k. 

Let  these  three  points  lie  on  the  circle  whose  equation  is 

then  (x-h-ay+{y-k-fiy=  B^, 

(x  +  h-ay  +  (y  +  k'  -  /8)2  =  i22. 

Subtracting  the  second  and  third  from  the  first, 

2h(x-cC)~h''  +  2k(iy-^-)-  A:2  =  0,| 
-2A(x-a)- A2_2A;'(y-/8)-A;'2  =  0,)  ^''^ 

whence  by  adding,  and  solving  for  y  —  /8, 

y-^=  2(k-k':,  •  ^^^ 

To  find  the  limit  of  this  fraction  as  A  ==  0,  let  y  =  0  (x)  bo 
the  equation  of  the  given  curve,  then 

y  -k  =  <\>(x  —  A),     y  +  y  —  <f>(x  +  K), 


258  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

wlience,  by  Taylor's  theorem, 

7(2 

y  +  k'=  4>(x)  +  h<l>'(x)  4- 1^  <\>"(x  +  dK), 
and  k  =  h<^'(x)  -  |^  </>"(a;  -  0^K),      . 

hence,  when  A  =  0, 

|  =  (/>'(:r),     |  =  f(a.),     ^  =  f'(^).  (4) 

Equation  (3)  may  now  be  written 


y-^  =  - 


k'  -k 


therefore,  by  (4), 


To  find  (a:  —  a),  divide  the  first  of  equations  (2)  by  2  A  and 
pass  to  the  limit,  then 


z  -  «  =  -  -  (2/  -  /3) 


.  ^\x')\\±\^{x)^_ 


da?' 

Thus  the  coordinates  («,  /3)  of  the  center  of  the  osculating 
circle  at  the  point  (x,  y)  are  the  same  by  either  definition. 
The  value  of  ^  is  then  found  as  before. 


150-151.] 


CONTACT  AND  CURVATURE 


259 


151.  Direction  of  radius  of  curvature.  Since  the  given 
curve  and  its  osculating  circle  at  a  point  P  have  the  same 
value  of  -^  at  that  point,  it  follows  that  they  have  the  same 

tangent  and  normal  at  P,  and  hence  that  the  radius  of 
curvature   coincides   with   the   normal.      Again,   since   the 

curve  and  its  osculating  circle  have  the  same  value  of  —^ 

dar 

at  P,  it  follows  from  Art.  141,  that  they  have  the  same 
direction  of  bending  at  that  point,  and  hence  that  the  center 
of  curvature  lies  on  the  concave  side  of  the  given  curve 
(Fig.  61). 

This  could  also  be  seen  from  the  fact  (Art.  150)  that  the 
osculating  circle  is  the  limiting  position  of  a  circle  passing 
through  three  points  on  the  curve  when  these  points  move 
into  coincidence. 


Fig.  61. 


Fig.  62. 


The  radius  of  curvature  is  usually  regarded  as  positive  or 
negative  according  as  the  bending  of  the  curve  is  positive 
or  negative  (Art.  141),  that  is,  according  as  the   value  of 


is  positive  or  negative ;   hence,  in  the  expression  for  J?, 


the   radical   in   the   numerator   is   always  to  be  given  the 
positive  sign.     The  sign  of  R  changes  as  the  point  P  passes 


DIFP.  CALC. 


18 


260  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

through  a  point  of  inflexion  on  the  given  curve  (Fig.  62). 
It  is  evident  from  the  figure  that  in  this  case  R  passes 
through  an  infinite  vahie ;  for  the  circle  through  the  points 
iV,  P,  Q  approaches  coincidence  with  the  inflexion  tangent 
when  JV and  ^approach  coincidence  with  JP;  and  thus  the 
center  of  this  circle  at  the  same  time  passes  to  infinity. 

152.  Other  forms  for  12. 

I.    Expression  for  M,  when  x  and  i/  are  functions  of  an 
independent  variable  t. 
By  Arts  21,  51, 


dy  fd^y     dx     d^x     dy' 

dy      dt      d^y      \d^  '  dt  ~  d¥  '  dt 


). 


dx      dx^     daP"  /'dx\^ 

dt  \dtj 

I'lierefore  the  expression  of  Art.  149  becomes 

<Py     dx  _  d^x     dy 
dt^  '  dt      di^  '  Tt 

II.    Expression  for  R,  when  the  curve  is  defined  by  an 
implicit  equation. 

Let  /(a;,  «/)=  0  be  the  equation  of  the  curve  ;  then  when 

the  value  of  -^,    —^  are  expressed  in  terms  of 
dx     dxr 

§f,    ^,    ^,     _^,    ^, 
dx     dy     dx^      dx  dy     dy^ 

(Ex.  10,  p.  182),  the  expression  for  Jl  becomes 

'dfV 

R  = 


KS)T 


dxj      \dy 


(df^dj      ^df  df    dj    ^  (dfVd'f 
\dyj  da?'         dx  dy  dxdy      \dxJ  dy^ 


151-154.]  CONTACT  AND  CURVATURE  261 

III.    Expression  for  R  in  polar  coordinates. 

If  the  equation  of  the  curve  be  given  in  the  form  p=zf(ff)^ 
the  expression  for  R  may  be  found  by  transforming  the 
equation  of  Art.  149,  by  means  of  the  relations 

x=p  cos  ^,     y=p  sin  0. 


The  result  is 


n-±^M 


.2       5V_,.,AapY 


153.  Total  curvature  of  a  given  arc;  average  curvature. 
The  total  curvature  of  an  arc  PQ  (Fig.  63)  in  which  the 
bending  is  continuous  and  in  one  direction,  is  the  angle 
through  which  the  tangent  swings  as 
the  point  of  contact  moves  from  the 
initial  point  P  to  the  terminal  point  Q ; 
or,  in  other  words,  it  is  the  angle 
between  the  tangents  at  P  and  Q^ 
measured  from  the  forward  end  of  the 
former  to  that  of  the  latter.     Thus  the 

total  curvature  of  a  given  arc  is  positive  or  negative  accord- 
ing as  the  bending  is  in  the  positive  or  negative  direction  of 
rotation. 

The  total  curvature  of  an  arc  divided  by  the  length  of  the 
arc  is  called  the  average  curvature  of  the  arc,  or  the  curva- 
ture for  unit  of  length.  Thus,  if  the  length  of  the  arc  PQ 
be  A»  centimeters,  and  if  its  total  curvature  be  A</>  radians, 

then  its  average  curvature  is  — ^  radians  per  centimeter. 

A« 

154.  Measure  of  curvature  at  a  given  point.  The  measure 
of  the  curvature  of  a  given  curve  at  a  given  point  P  is  the 


262 


DIFFERENTIAL   CALCULUS 


[Cii.  XVI. 


limit  which  the  average  curvature  of  the  arc  PQ  approaches 
when  the  point  Q  approaches  coincidence  with  P. 

Since  the  average  curvature  of   the  arc  PQ  in  —2,  the 
measure  of  the  curvature  at  the  point  P  is 


K  = 


lim     A^  _  d^ 


and  may  be  regarded  as  the  rate  of  deflection  of  the  arc  from 
the  tangent  estimated  per  unit  of  length  ;  or  again,  as  the 
ratio  of  the  angular  velocity  of  the  tangent  to  the  linear 
velocity  of  the  point  of  contact. 

To  express  k  in  terms  of  x,  y,  and  their  derivatives.     Since 


then 


and 


tan  0  =  -r^i 
ax 

<f>  =  tan 


-1^ 
dx 


#^A(^tan-i^^ 
ds      ds 


=  Aftan-i^^  •  — 
dx\  dxj     ds 


dx^ 


1  _i_  (^y^  c?8 

\dxJ      dx 


therefore 


ds 


^y 

dx^ 


2)1 


'<Wi 


[Art.  121 


155.  Curvature  of  an  arc  of  a  circle.     In  the  case  of  a  cir- 
cular arc  the  normals  are  radii ; 


154-156. j  CONTACT  AND   CURVATURE  263 

hence  A«  =  r  •  A<f>,    -^  =  -,  (1) 

As      r  ^ 

thus  K  =  — 

r 

Thus  the  average  curvature  of  all  arcs  of  the  same  circle 
is  constant  and  equal  to  -  radians  per  unit  of  length. 

For  example,  in  a  circle  of  2  feet  radius  the  total  curva- 
ture of  an  arc  of  3  feet  is  |  =  1.5  radians,  and  the  average 
curvature  is  .5  radian  per  foot. 

It  also  follows  from  (1)  that  in  different  circles,  arcs  of 
the  same  length  have  a  total  curvature  inversely  propor- 
tional to  their  radii. 

Thus  on  a  circumference  of  1  meter  radius,  an  arc  of  5  decimeters  has 
a  total  curvature  of  .5  radian,  and  an  average  curvature  of  .1  radian  per 
decimeter ;  but  on  a  circumference  of  half  a  meter  radius,  the  same  length 
of  arc  has  a  total  curvature  of  1  radian  and  an  average  curvature  of  .2 
radian  per  decimeter. 

156.  Curvature  of  osculating  circle.  A  curve  and  its  oscu- 
lating circle  at  P  have  the  same  measure  of  curvature  at  that 
point. 

For,  let  K,  k'  be  their  respective  measures  of  curvature  at 
the  point  of  contact  (x^  y) ;  then  from  Art.  154, 

K  = « 

\2lf 


and  from  Art.  149, 


V  -m 


^y 

,1                d:i^  ,  , 

k'=-  = T,    lience  k  =  k' . 


264  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

It  is  on  account  of  this  property  that  the  osculating  circle 
is  called  the  circle  of  curvature.  This  is  sometimes  used  as 
the  defining  property  of  the  circle  of  curvature.  The  radius 
of  curvature  at  P  would  then  be  defined  as  the  radius  of  the 
circle,  whose  measure  of  curvature  is  the  same  as  that  of  the 
given  curve  at  the  point  P.  Its  value,  as  found  from  Art. 
154  and  Art.  155,  accords  with  that  given  in  Art.  149. 

EXERCISES 

1.  Find  the  order  of  contact  of  the  two  curves 

y  =  x^,     y  =  3x^  —  Sx  +  I. 

2.  Find  the  order  of  contact  of  the  parabola  y^  =  ^x,  and  the 
straight  line  3  y  =  x  +  9. 

3.  Find  the  order  of  contact  of 

9  ?/  =  x3- 3x2  +  27    and    9?/  +  3x  =  28. 

4.  Find  the  order  of  contact  of 

y  =  log (x  -  1)     and     x^-6x  +  2y  +  8  =  0  at  (2,  0). 

5.  Show  that  the   circle  {x-—\\(y---Y=^-^  and  the  curve 

Vx  +  Vy  =;  Va  have  contact  of  tlie  third  order  at  the  point  x  =  y  =  -• 

4 

6.  What  must  be  the  value  of  a  in  order  that  the  parabola 

y  =  X  +  1  +  r;  (x  —  1)'^ 
may  have  contact  of  the  second  order  with  the  hyperbola  x^  =  3  x  —  1  ? 

7.  Find  the  order  of  contact  of  the  parabola 

(x-2ay  +  (y-2ay  =  2xy, 
and  the  hyperbola  xy  =  a^. 

EXERCISES    ON    CURVATURE 

8.  Tn  the  curve  _y  =  x*  —  4x^  —  ISx^,  the  radius  of  curvature  at  the 
origin  is  ^V- 

9.  Show  that  the  two  radii  of  curvature  of  the  curve 

y^  =  x^  '  

n  —  X 

at  the  origin  are  ±  a  V2;    and  that    R  =  I  a  at  (—  a,  0). 


166-157.] 


CONTACT  AND  CURVATURE 


265 


find  the  radius  of  curvature  in  each  of  the  following  curves : 

10.   The  parabola 

y^  =  i  ax. 

11.   The  ellipse 

a2  "^  62  -  ^• 

12.  The  catenary 

y- 

=  £(^ic  +  e% 

13.  The  exponential  curve     y  =  ae' 

14.  The  parabola  Vx  +  y/y  =  2  v'a. 

15.  The  hypocycloid  xs-\-yi=z  a*. 

16.  The  curve  y  =  log  sec  a:.    Catenary  of  uniform  strength. 

17.  Derive  the  formula  —  =  f^V+  (^Y. 

_=cos<A;    _  =  -sm<^.^  =  --^;ete.J 


157.  Direct  derivation  of  the  expressions  for  k  and  B  in 
polar  coordinates. 

Using  the  notation  of  Art.  119, 


hence 


^d(f>^dd  _\    '^  de) 


ds 
dd 

ds 
dd 

A'* 

d^\ 

dej 

^-8)7 


de 


But  tan-^  = />— -,     -^  =  tan~i 
dp 


dp 
10 


CI) 


[Art.  124 


266  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

therefore,  by  differentiating  as  to  d  and  reducing, 

djrjKdd)      Pdd"^ 
which,  substituted  in  (1),  gives 


P^-  P: 


["^mi 


and  the  relation  i2  =  -  then  reproduces  the  result  obtained 

K 

in  Art.  152  by  transformation  of  coordinates. 

When  u=-  is  taken  as  dependent   variable,  the  expres- 
P 
sion  for  k  assumes  the  simpler  form 

K  = 


Since  at  a  point  of  inflexion  k  vanishes  and  changes  sign, 

hence  the  condition  for  a  point  of   inflexion,  expressed  in 

d^u 
polar  coordinates,  is  that  u  +  ^j^  shall  pass  through  zero 

and  change  its  sign.     See  Art.  144. 


EXERCISES 

1.  Show  that  the  radius  of  curvature  of  the  curve 

p  =  a  sin  nd  at  (0,  0)  is  — • 

2.  Find  the  radius  of  curvature  of  p"  =  a"  cos  m6. 
Find  the  value  of  R  in  each  of  the  following  curves : 

3.  The  circle  p  =  a  sin  6. 


157-168.]  CONTACT  AND  CURVATURE  267 

4.  The  lemniscate  p^  =  a^  cos  2  6. 

5.  The  logarithmic  spiral  p  =  e"^. 

6.  The  trisectrix  p  =  2  a  cos  6  —  a. 

7.  The  equilateral  hyperbola  p^  cos  2  6  =  a\ 

8.  For  any  curve  prove  the  formula 

R  = 


EVOLUTES   AND  INVOLUTES 

158.  Definition  of  an  evolute.  When  the  point  P  moves 
along  the  given  curve,  the  center  of  curvature  0  describes 
another  curve  which  is  called  the  evolute  of  the  first. 

Let  /(a:,  y)  =  0  be  the  equation  of  the  given  curve,  then 
the  equation  of  the  locus  described  by  the  point  0  is  found 
by  eliminating  x  and  t/  from  the  three  equations 


X—  a  = 


di/ 
dx 


'<%. 


m 


daP' 


dx^ 

and  thus  obtaining  a  relation  between  a,  ^,  the  coordinates 
of  the  center  of  curvature. 

No  general  process  of  elimination  can  be  given  ;  the 
method  to  be  adopted  depends  upon  the  form  of  the  given 
equation /(a;,  y)  =  0. 


268 


DIFFERENTIAL  CALCULUS 


[Ch.  XVI. 


Ex.  1.    Find  the  evolute  of  the  parabola  y^  =  4  px. 


Since 


ri    '11  =  nh 


(Py 


y  =  2pixi,    '-^=p^x  2, 

ax  dx^ 


\p^^-K 


hence  x  —  a  =  —  p^x  2(1+  px  ^)  2p  2x5  =  _  2  (x  +  />), 

and  1/  —  /3  =  (1  -^  px~^)  2/)~"2a;2  =  2  (/j~^a;2  +  p'2x'i) ; 


therefore 


a  =  2/)  +  3  a;, 


P  --  2/>~ix2, 


Fio.  64. 


when,  by  eliminating  x,   ^  (a  —  2/))'  =  \  (p^fi)% 

i.e.,  4  (a  -  2 py  =  27 p^, 

is  the  equation  of  the  evolute  of  th^  parabola,  in  which  a,  yS  are  current 
coordinates. 


Ex.  2.    Find  the  evolute  of  the  ellipse 


(1) 


158-159.]  CONTACT  AND  CURVATURE  269 

Here  E  +  L.^  =  o,     ^  =  _^ 

a^     b^    dx  dx         ahf 


y  -X 


dy 


whence 

^-^  =  — ^*-^-(^  +  a^^)^=(^+l-p>• 
The^ef  ore  -  )8  =  ^-^  y*.  (2) 

Similarly,  a  = —  3*.  (3) 

Eliminating  x,  y  between  (1),  (2),  (3),  the  equation  of  the  locus  de- 
scribed by  (a,  (i)  is 

(aa)^  +  (6)3)^  =  (a2  -  b'^)i.  (Fig.  69.) 


159.  Properties  of  the  evolute.  The  evolute  has  two  im- 
portant properties  that  will  now  be  established. 

I.  The  normal  to  the  curve  is  tangent  to  the  evolute. 
The  relations  connecting  the  coordinates  (a,  /9)  of  the  center 
of  curvature  with  the  coordinates  (a:,  y)  of  the  correspond- 
ing point  on  the  curve  are,  by  Art.  149, 

«-«  +  (y-/3)g=0,  (1) 

From  these  equations  «,  /3  may  be  considered  functions  of 
X ;  hence,  b}'  differentiating  (1),  regarding  «,  ^,  i/  as  func- 
tions of  a;. 


270  DIFFERENTIAL   CALCULUS 

Subtracting  (3)  from  (2)  gives 

da      d^  ^y  _  (\ 
dx     dx  dx        ' 


[Ch.  XVI. 


(4) 


whence 


d^ 


dS 
da 


dx 
dy 


but-  -T-  is  the  slope  of  the  tangent  to  the  evolute  at  («,  /S); 
and  —  —  is  the  slope  of  the  normal  to  the  given  curve  at 

(a;,  y).  Hence  these  lines  have  the  same  slope ;  but  they 
pass  through  the  same  point  («,  y8), 
therefore  they  are  coincident. 

II.  The  difference  between  two 
radii  of  curvature  of  the  given  curve, 
touching  the  evolute  at  the  points 
(7i,  C^  (Fig.  65),  is  equal  to  the  arc 
C^C^  of  the  evolute. 

Since  R  is  the  distance  between 
points  (x^  ?/),  («,  yS),  hence 


Fie.  66. 


(a;-«)2  +  (y-y3)2=i22. 


(5) 


When  the  point  (a;,  y')  moves  along  the  given  curve,  the 
point  (a,  y8)  moves  along  the  evolute,  and  thus  a,  /8,  72,  y 
are  all  functions  of  x. 

Differentiation  of  (5)  as  to  x  gives 

hence,  subtracting  (6)  from  (1), 


,  ,da.        ^^  d3  -ndR 


dx 


dx 


(7) 


159.]  CONTACT  AND  CURVATURE  271 

Again,  from  (1)  and  (4), 

da  dl3 


dx  dy 


X- a      y- fi 
Hence,  each  of  these  fractions  is  equal  to 


(8) 


< 


daV_^(d^?  da 

dxj       \dzJ  dx  ^Q^ 


^Qx-ay-hiiy-^y      R' 

in  which  a  is  the  arc  of  the  e volute. 

Next,  multiplying  numerator  and  denominator  of  the  first 
member  of  (8)  by  a;  —  a,  and  those  of  the  second  member  by 
y  —  p.  and  combining  new  numerators  and  denominators,  it 
follows  that  each  of  the  fractions  in  (8)  is  equal  to 


(^ 

-ay+iy-^y 

which  equals 

RdR 

by  (7) 

and  (5). 

Whence,  by  (9), 

d(T  _ 
dx 

_^dR 

dx^ 

that  is. 

i2)=0: 

therefore 

a 

■±R  = 

constant. 

(10) 

wherein  a  is  measured  from  a  fixed  point  A  on  the  evolute. 
Now,  let  Cj,  Cj  be  the  centers  of  curvature  for  the  points 
Pj,  Pg  on  the  given  curve  ;  let  P^Cj  =  Rj^,  P'S'i  =  R^  \  and 
let  the  arcs  A  (7j,  A  C^  be  denoted  by  o-j,  a^  ;  then 

o-j  ±  iJj  =  0-2  ±  i^a'  by  (10); 


272 


DIFFERENTIAL    CALCULUS 


[Ch.  XVI. 


that  is, 
hence, 


o-j  —  0-2  =  ±  (R^  —  i^i); 
arc  Cj  Cg  =  R^ 


R,. 


(11) 


Fi8.  66. 


Thus,  in  figure  66, 

-^2^2  ^"   ^2^3  ~  -^3^3'  ®tC* 

Hence,  if  a  thread  be  wrapped 
around  the  evolute,  and  then  be 
unwound,  the  free  end  of  it  can 
be  made  to  trace  out  the  original 
curve.  From  this  property  the 
locus  of  the  center  of  curvature 
of  a  given  curve  is  called  the 
and  the  latter  is  called  the  involute 


evolute  of  that  curve 
of  the  former. 

When  the  string  is  unwound,  each  point  of  it  describes  a 
different  involute  ;  hence  to  the  same  curve  correspond  an 
infinite  number  of  involutes  ;  but  a  curve  has  but  one 
evolute. 

Any  two  of  these  involutes  intercept  a  constant  distance 
on  their  common  normal,  and  are  called  parallel  curves. 

EXERCISES 
Find  the  coordinates  of  the  center  of  curvature  of  the  following 
curves : 

1.  The  parabola  y^  =  4  ax. 

2.  The  semicubical  parabola  a;'  =  ay^. 

3.  The  four-cusped  hypocycloid  x^  +  y^  =  a* 


5.   The  equation  of  the  equilateral  hyperbola  being  xy  =  a^,  prove 
that 

„     a(a     xy  „      a/a      xV 

and  derive  the  equation  of  the  evolute. 


159.] 


CONTACT  AND   CURVATURE 


273 


6.  Show  that  the  curvature  of  an  ellipse  is  a  minimum  at  the  end  of 
the  minor  axis,  and  that  the  osculating  circle  at  this  point  has  contact 
of  the  third  order  with  the  curve; 


Fig.  6T. 

This  circle  of  curvature  must  be  entirely  outside  the  ellipse  (Fig.  67); 
for,  consider  two  points  P■^,  P^  one  on  each  side  of  B,  the  end  of  the 
minor  axis.  At  these  points  the  curvature  is  greater  than  at  B,  hence 
these  points  must  be  farther  from  the  tangent  at  B  than  the  circle  of 
curvature,  which  has  everywhere  the  same  curvature  as  at  B. 

7.  Similarly,  show  that  the  curvature  at  A,  the  end  of  the  major 
axis,  is  a  maximum,  and  that  the  circle  of  curvature  at  A  lies  entirely 
within  the  ellipse  (Fig.  67). 

8.  Show  how  to  sketch  the  circle  of  curvature  for  points  between  A 
and  B.  The  circle  of  curvature  for  points  between  A  and  B  has  three 
coincident  points  in  common  with  the  ellipse  (Art.  150),  hence  the  circle 


Fie.  6a 


274 


DIFFERENTIAL   CALCULUS 


[Ch.  XVI. 


crosses  the  curve  (Art.  147).  Let  K,  P,  L  be  three  points  on  the  arc,  such 
that  K  is  nearest  A,  L  nearest  B.  The  center  of  cui'vature  for  P  lies  on 
the  normal  to  P,  and  on  the  concave  side  of  the  curve,  'i'he  circle  crosses 
at  P,  lying  outside  of  the  ellipse  at  K  (on  the  side  towards  A),  and 
inside  the  ellipse  at  L ;  for  the  bending  of  the  ellipse  increases  from  B 
to  P  and  from  P  to  K,  while  the  bending  (curvature)  of  the  osculating 
circle  remains  constant  (Fig.  68). 

9.    Two  centers  of  curvature  lie  on  every  normal ;  prove  geometrically 
that  the  normals  to  the  curve  are  tangents  to  the  evolute. 

10.  Show  that  the  entire  length  of  the   evolute  of   the  ellipse  is 

—  )•     [From  equation  (11)  above,  take  /?,,  i?,  as  the  radii  of 

b       a  J 
curvature  at  the  extremities  of  the  major  and  minor  axes.] 

11.  Show  that  in  the  parabola  y^  =  4:ax  the  length  of  the  part  of  the 
evolute  intercepted  within  the  parabola  is  4a(;iVl}  _  1). 

12.  Show  that  in  the  parabola  Vx  +  Vy=  Va  the  relation  exists, 
a  +  )3  =  3  (a;  +  y). 

13.  If  E  be  the  center 
of  curvature   at  the  vertex 
~  A    (Fig.    69),    prove    that 

CE  =  ae%  in  which  e  is  the 
eccentricity  of  the  ellipse; 
and  hence  that  CD,  CA, 
CF,  CE  form  a  geometric 
series  whose  common  ratio 
is  e.  Show  also  that  DA, 
AF,  FE  form  a  similar 
series. 


D 


aI  FE^ 

B  \ 

_/J 

/ 

Fig, 


m 

69. 


14.  If  H  be  the  center  of 
curvature  at  B,  show  that 
the  point  H  is  without  or 
within  the  ellipse,  accoiding 
as  a  >  or  <  bV2,  or  accord- 
ing as  e=^  >  or  <  J. 

15.  Show  by  inspection 
of  the  figure  that  four  leal 
normals  can  be  drawn  to  the 
ellipse  from  any  point  within 
the  evolute. 


CHAPTER   XVII 
SINGULAR  POINTS 

160.  Definition    of    a    singular    point.      If    the    equation 

dv 
f(^x,y~)=0  be  represented  by  a  curve,  the  derivative  -^, 

when  it  has  a  determinate  value,  expresses  the  slope  of  the 
tangent  at  the  point  {x,  y).  There  may  be  certain  points  on 
the  curve,  however,  at  which  the  expression  for  the  deriva- 
tive assumes  an  illusory  or  indeterminate  form  ;  and,  in 
consequence,  any  line  whatever  drawn  through  such  a  point 
may  be  regarded  as  a  tangent  at  the  point.  Such  values  of 
a;,  y  are  called  singular  values^  and  the  corresponding  points 
on  the  curve  are  called  singular  points. 

161.  Determination  of  singular  points  of  algebraic  curves. 
When  the  equation  of  the  curve  is  rationalized  and  cleared 
of  fractions,  let  it  take  the  form  /(a;,  «/)=  0. 

This  gives,  by  differentiation  with  regard  to  a;,  as  in 
Art.  96, 

dx      dy  dx 

dii  dx 

whence  dx  =  -df  ^^^ 


dy_ 


by 


In  order   that  -^  may  become   illusory,  it  is  therefore 
dx 

necessary  that  -ir=^-^  -^r  =  ^-  (2) 

ax  ay 

DIFF.  CALC.  — 19  275 


276  DIFFERENTIAL    CALCULUS  [Ch.  XVII. 

Thus  to  determine  whether  a  given  curve  /  (2;,  ?/)  =  0  has 

singular  points,  put  -^  and  -^  each  equal  to  zero  and  solve 
ox  ay 

these  equations  for  x  and  y. 

If  any  pair  of  values  of  x  and  y^  so  found,  satisfy  the 
equation  /  (aj,  ^)  =  0,  the  point  thus  determined  is  a  singular 
point  on  the  curve. 

To  determine  the  appearance  of  the  curve  in  the  vicinity 
of  a  singular  point,  (x^,  y^  evaluate  the  indeterminate  form 

dy  _     dx  _0 

dy 

by  finding  the  limit  approached  continuously  by  the  slope  of 
the  tangent  when  x  =  Xy,  y  =  yj, 

Am 

thus  §1  dx\dx) 

dx  d^fd£ 

dx\dy 

gy  I     d^f    dy 
_  _  dx^     dxdydx  r^^^^  72,  96. 

~     ay     52/  dy 

dxdy      dy^  dx 

This  equation  cleared  of  fractions  gives,  to  determine  the 
slope  at  (a^j,  y{),  the  quadratic 

^W+2-^('^V^  =  0.  (3) 

dy"-  \dxj  dx  dy  \dxj      dofi 

This  quadratic  equation  has  in  general  two  roots.  The 
only  exception  is  when  simultaneously,  at  the  point  in 
question, 

^-0      ^'Z_-o     ^-0  a-) 

dx'-^'      dxdy-^'      6/-"'  ^  -^ 


161-162.]  SINGULAR   POINTS  277 

in  which  case  -=^  is  still  indeterminate  in  form,  and  must  be 
ax 

evaluated  as  before.     The  result  of  the  next  evaluation  is  a 

cubic  in  -^,  which  gives  three  values  of  the  slope,  unless  all 

the  third  partial  derivatives  vanish  simultaneously  at  the 
point. 

The  geometric  interpretation  of  the  two  roots  of  equation 
(3)  Avill  now  be  given,  and  similar  principles  will  apply 
when  the  quadratic  is  replaced  by  an  equation  of  higher 
degree. 

The  two  roots  uf  (3)  are  real  and  distinct,  real  and  coin- 
cident, or  imaginary,  according  as 

\dxdyj       doc^  dy^ 

is  positive,  zero,  or  negative.  These  three  cases  will  be  con- 
sidered separately. 

162.   Multiple  points.     First  let  H  be  positive.     Then  at 

the  point  (a-,  ^)  for  which  -^^  =  0,  -^  =  0,  there  are  tWo  values 

dx  by 

of  the  slope,  and  hence  two  distinct  singular  tangents ; 
thus  the  curve  goes  through  the  point  in  two  directions, 
or,  in  other  words,  two  branches  of  the  curve  cross  at  this 
point.  Such  a  point  is  called  a  real  double  point  of  the 
curve,  or  simply  a  node.  The  conditions,  then,  to  be  satisfied 
at  a  node  (a^j,  y-^  are  that 

•^  ^  ^  "^^^  bx^  dy^ 

and  that  -ff^ajj,  ^j)  be  positive. 

Ex.     Examine  for  singular  points  the  curve 

3a;2  -  a:^  -  2y2  +  x8  -  8/  =  0. 


278 


DIFFERENTIAL   CALCULUS 


[Ch.  XVII. 


Here 


dx  •"  By 


a:  —  4  3/  —  24  ^^ 


The  values   x  =  0,   y  =  0   will  satisfy  these  three  equations,  hence 
(0,  0)  is  a  singular  point. 


Since 


||=6  +  6x  =  G  at  (0,0), 
^  =  -l=-lat(0,0), 


ay 


=  _4  _48y  =  -4  at  (0,0), 


hence  the  equation  of  the  slope  is,  from  (8), 

-*(iy-Ki)+«=»- 

of  which  the  roots  are  1  and  —  |.     Thus  (0,  0)  is  a  double  point  at  which 
the  tangents  have  the  slopes  1,  —  f. 


Fig.  to. 


163.  Cusps.  Next  let  11=  0.  The  two  tangents  are  then 
coincident,  and  there  are  two  cases  to  consider.  If  the  curve 
recedes  from  the  tangent  in  both  directions  from  the  point 
of  tangency,  it  is  called  a  point  of  osculation;  and  two 
branches  of  the  curve  touch  each  other  at  this  point.      If 


162-163.]  SINGULAR  POINTS  279 

both  branches  of  the  curve  recede  from  the  tangent  in  only 
one  direction  from  the  point  of  tangency,  the  point  is  called 
a  cusp. 

Here  again  there  are  two  cases  to  be  distinguished.  If 
the  brandies  recede  from  the  point  on  opposite  sides  of  the 
double  tangent,  the  cusp  is  said  to  be  of  the  first  kind ;  if 
the,y  recede  on  the  same  side,  it  is  called  a  cusp  of  the  second 
kind. 

The  method  of  investigation  will  be  illustrated  by  a  few 
examples. 

Ex.  1.  /(a:,  y)  =  aY  -  a^  +  x"  =  0. 

|^=-4a%;8  +  6x«;     ^=2aV 
dx  _  '    dy  " 

The  point  (0,  0)  will  satisfy  /(x,  y)=  0,  -^  =  0, -^  -  0;  hence  it  is  a 
singular  point.     Proceeding  to  the  second  derivatives, 

^  =  -  12  a!^2  +  30x*  =  0  at  (0,  0), 


dxdy 


The   two  values   of   -^  are  therefore  coincident,  and  each  equal  to 
dx 

zero.  From  the  form  of  the  equation,  the  curve  is  evidently  symmet- 
rical with  regard  to  both  axes;  hence  the  point  (0,  0)  is  a  point  of 
osculation. 

No  part  of  the  curve  can  be  at  a  greater  distance  from  the  ^/-axis 

than  ±  a,  at  which  points  ^  is  infinite.     The  maximum  value  of  y 
dx 

corresponds  to  x  =  ±aV\.     Between  a;  =  0,  x  =  ay/\  there  is  a  point 

of  inflexion  (Fig.  71). 


280 


DlFfERBN TlAL   CALCUL  tiS 


[Ch.  xvii. 


Ex.2.  f(x,,j)=y^-x^  =  0; 

Hence  the  point  (0,  0)  is  a  singu- 
lar point. 


Again, 


dx' 


-  6  X  =  0  at  (0,  0) ; 


Fig.  71. 


dxdy 


=  0; 


ay 


=  2. 


dx 


Therefore  the  two  roots  of  the  quadratic  equation  defining  —  are  both 

dy 
equal  to  zero.     Thus  far,  this  case  is  exactly  like  the  last  one,  but  here 
no  part  of  the  curve  lies  to  the  left  of  the  axis  of  y.     On  the  right  side, 
the  curve  is  symmetric  with  regard  to  the  x-axis.     As  x  increases,  y  in- 
creases ;  there  are  no  maxima  nor  minima,  and  no  inflexions  (Fig.  72). 

Ex.  3.  y"(x,  ^)  =  x*  —  2  ax'^y  —  axy^  +  a^y"^  =  0. 

The  point  (0, 0)  is  a  singular  point,  and  the  roots  of  the  quadratic 

defining  -2  are  both  •equal  to  zero. 
dx 


Let  a  be  positive.     Solving  the  equation  for  y, 
a  —  x\  'a/ 


When  X  is  negative,  y  is  imaginary ;  when  x  =  0,  ^  =  0;  when  x  is 
positive,  but  less  than  a,  y  has  two  positive  values,  therefore  two  branches 


Fig.  72. 


Fig.  73. 


are  above  the  x-axis.  When  x  =  a,  one  branch  becomes  infinite,  having 
the  asymptote  x  =  a;  the  other  branch  has  the  ordinate  J  a.  The  origin 
is  therefore  a  cusp  of  the  second  kind  (Fig.  73). 


163-164] 


SINGULAR  POINTS 


281 


164.  Conjugate  points.  Lastly,  let  ff  be  negative.  In 
this  case  there  are  no  real  tangents ;  hence  at  the  point 
in  question,  no  points  in  the  immediate  vicinity  of  the 
given  point  satisfy  the  equation  of  the  curve. 

Such  an  isolated  point  is  called  a  conjugate  point. 

Ex.  /(z,  y)  =  ay^  -  z»  +  bx^  =  0. 

Here  (0,  0)  is  a  singular  p6int  of  the  locus,  and 

ax  a 

both  roots  being  imaginary  if  a  and  6 
have  the  same  sign. 

To  show  the  form  of  the  curve,  solve 
the  given  equation  for  y, 


then 


y  =  ±  x-^- 


hence,  if  a  and  b  are  positive,  there  are 
no  real  points  on  the  curve  between  x=0 
and  x  —  b.  Thus  O  is  an  isolated  point 
(Fig.  74). 


Fig.  74. 


These  are  all  the  singularities  that  algebraic  curves  can 
have,  though  complicated  combinations  of  them  may  appear. 
In  all  the  foregoing  examples,  the  singular  point  was  (0,  0); 
but  for  any  other  point,  the  same  reasoning  will  apply. 


Ex. 


/(ar,  y)  =  a;2  +  3  /  -  13  //2  -ix  +  l7y-B=0, 
^  =  2x-4,       §£  =  9w2_26y +  17. 


df. 


dx 


0,  ^  =  0;    hence  (2,  1)  is  a 


dy 


At  the  point  (2,1),  /(2,   1)  =  0, 
singular  point. 

Also       ^  =  2;     -^  =  0;     ^=  18y  -  26,  =  -  8  at  (2,  1). 
dx^  dxdy  dy^         "  ^       ^ 

Hence  -^=  ±  2;  and  thus  the  equations  of  the  two  tangents  at  the 
fix 

node  (2,  1)  are    y  -\  =  2{x-2),    y  -  1  =  -  2(x  -  2). 


282  DIFFERENTIAL   CALVULUS        [Ch.  XVII.  164. 

When,  at  a  singular  point,  H  is  negative,  the  point  is 
necessarily  a  conjugate  point,  but  the  converse  is  not  always 
true.  A  singular  point  may  be  a  conjugate  point  when 
H=Q  (cf.  Ex.  9). 

Transcendental  singularities.  A  curve  whose  equation  involves  a 
transcendental  function  may  have  a  stop-point,  at  which  the  curve  ter- 
minates abruptly  (Fig.  48)  or  a  salient  point  at  which  two  branches  of 
the  curve  meet  and  stopwdthout  having  a  common  tangent.  In  the  first 
case  there  is  a  discontinuity  in  the  function;  in  the  second,  a  discon- 
tinuity in  the  derivative. 

They  are  usually  discovered  by  inspection  in  tracing  the  curve. 

EXERCISES 

Find  the  multiple  points,  and  tlie  direction  of  the  tangents  at  them, 
in  the  following  curves : 

1.  «2^/2  =  a2j.2  _  4  a.,3  3.    (a;2  ^  y^y  =  i^  aH'^y\ 

2.  a;*  -  2  a^3  _  3  „2y2  _  2  „2^2 + ^4  _  q.       4.    if  =  x'^  ^-  x\ 

5.  If  a^2  —^j,  _  (xy(^x  —  ft),  show  that,  when  x  =a,  there  is  a  con- 
jugate point  if  a  be  less  than  b,  a  double  point  if  a  be  greater  than 
h,  and  a  cusp  if  a  be  equal  to  h. 

6.  Show  that  the  curve  y^  =  (^x  —  a)\x  —  c)  has  a  cusp  of  the  first 
kind. 

7.  Draw  the  curve  x^  -]-  y^  =  x^  +  y^  in  the  vicinity  of  the  origin. 

8.  Prove  that  the  curve  x*  —  2  ax^y  —  axif  +  ahf  =  0  has  a  cusp  of 
the  second  kind  at  the  origin. 

9.  What  change  in  the  coefficient  of  xhj  in  the  last  example  will 
make  the  origin  a  conjugate  point?  Show  that  the  tangents  at  this 
point  are  still  real  and  coincident. 

10.  Trace  the  curve  x*  +  2  axhj  —  ay^  =  0  for  points  near  the  origin. 

1 

11.  In  the  curve  y(\  +  e*)  =  x,  show  that  if  x  =  0  from  positive  side, 

y  .      •  y 

-  =  0 ;  if  from  negative  side,  -  =  1 ;  hence  a  discontinuity  in  slope,  i.e., 
a  salient  point. 


CHAPTER   XVIII 

CURVE  TRACING 

165.  Traciug  a  curve  consists  in  finding  its  general  form 
when  its  equation  is  given. 

Three  kinds  of  equations  present  themselves. 

1.  Cartesian  equations  ; 
(a)  algebraic  ; 

(6)  transcendental. 

2.  Polar  equations. 

There  is  no  fixed  method  of  procedure  applicable  to  all 
cases.  A  few  general  suggestions  for  Cartesian  equations 
will  be  given,  and  then  some  examples  worked  out  in  detail. 

Find   -^  ;    this  will  give  the  direction  of  the  curve  at 
ax 

any  point,  and  will  serve  to  locate  maximum  and  minimum 
ordinates. 

Examine  for  asymptotes,  and  construct  them.  Deter- 
mine on  which  side  of  each  asymptote  the  corresponding 
infinite  branch  is  situated. 

Find  —^  ;  this  will  give  the  direction  of  bending  at  any 

point,  and  will  determine  the  points  of  inflexion. 

Examine  algebraic  curves  for  singular  points,  and  deter- 
mine whether  they  are  nodes,  cusps,  or  conjugate  points. 

If  the  minute  configuration  of  a  curve  at  any  particular 
point  is  desired,  it  is  often  expeditious   to  transform   the 

283 


284  DIFFERENTIAL   CALVULUS  [Ch.  XV III. 

origin  to  that  point,  and  then  neglect  the  higher  powers  of  x 
and  y,  as  relatively  unimportant.  This  principle  will  be 
used  and  discussed  in  some  of  the  examples  that  follow. 

166.  Illustration.     Trace  the  curve 

This  curve  goes  through  the  origin  ;  and  it  is  symmetric 
with  regard  to  the  a;-axis,  for  the  equation  is  not  changed 
when  y  is  changed  to  —  «/ ;  but  it  is  not  symmetric  with 
regard  to  the  «/-axis. 

Putting  a;  =  0  gives  ^  =  0 ;  and  putting  y  =  0  gives 
a:*  =  0 ;  hence  the  curve  does  not  intersect  either  of  the 
coordinate  axes,  except  at  the  origin. 

Since        ^^43^  +  6i/\     ^  =  -4f  +  12xi/. 


hence 


dy  2^:3  +  3^2 


dx  (Q  X  —  2  y^^y 

which  becomes  indeterminate  only  for  a;  =  0,  y  =  0. 

Thus  the  origin  is  a  singular  point  of  the  curve.     The 
second  partial  derivatives  are 

i=-^-  ^.-^^'   i=-- 

which  all  vanish  at  the  origin ;    hence  those  of  the  third 
order  must  also  be  obtained  : 

dx^  '      dx^dy         '     dxdy^  '     dy^ 

dv 
The  general  equation  determining  -^,  derived  similarly 

to  that  in  Art.  149,  is 

dy^\dx)         dxdy'^\dxj         dxdy\dx)     da^        ' 


165-16G.]  CURVE  TliACiNQ  285 

which  becomes  in  this  case 

Thus  two  values  of  -^  are  0,  and  the  third  root  is  infinite  ; 

showing  that  the  a;-axis  is  tangent  to  two  branches,  and  the 
y-axis  to  a  third  branch. 

To  obtain  the  form  of  the  first  branches  in  the  vicinity  of 
the  origin  it  may  be  observed  that  since  on  these  branches 
1/  is  evidently  an  infinitesimal  of  a  higher  order  than  a;, 
hence  «^  may  be  neglected  in  comparison  with  the  other 
terms,  and  there  results  a:^  =  —  6  ?/'-*,  as  the  equation  of  a 
curve  approximately  coinciding  with  the  two  branches  in 
question  near  the  origin. 

This  curve,  and  hence  also  the  given  curve,  has  obviously 
a  cusp  of  the  first  kind  lying  to  the  left  of  the  axis  of  y. 

Similarly,  in  the  case  of  the  branch  that  is  tangent  to  the 
y-axis,  a^  may  be  neglected,  and  the  resulting  curve  is 
y^z=  Qx,  which  is  a  parabola  situated  on  the  right  side  of 
the  y-axis. 

Thus,  the  third  branch  is  parabolic  in  form  near  the 
origin. 

By  solving  for  y, 


y 


=  ±V3a;-f  \/9a,-2  +  a;4, 


in  which  only  the  positive  sign  is  to  be  retained  before  the 
inner  radical,  as  the  negative  sign  would  give  imaginary 
values  to  y.  Any  line  parallel  to  the  y-axis  will  therefore 
meet  the  curve  in  only  two  points. 

Again,  regarding  y  as  given,  the  resulting  equation  in  x 
has  one  positive  root  between  0  and  y  because  /(O,  i/)  is 
negative,  and/(^,  y)  is  positive,  and  similarly  one  negative 


286  DIFFERENTIAL  CALCULUS  [Ch.  XVIII. 

root  numerically  greater  than  y ;  the  others  being  imagi- 
nary. Thus  no  branch  of  the  curve  crosses  the  lines  x=  ±i/, 
except  at  the  origin. 

To  examine  for  asymptotes.     Put  y  =  mx  +  J,  then 

a^  —  {vix  +  i)*  +  6  x(mx  +  h)^  =  0  ; 

I.e.,     (1  —  w!^')3^  +  (—  4  vrfib  +  6  m^')3^  +  (—  6  rn^h^  +  12  mb)3^ 

Let  l-w4  =  0    and    -4^3^  +  6^2  =  0, 

then  w  =  ±  1,        6  =  ±  |, 

thus  y  —  x+^,     y  =  —  x  —  ^ 

are  the  asymptotes.  The  other  two  asymptotes  are  imagi- 
nary. 

To  find  the  finite  points  in  which  this  asymptote  cuts  the 
curve,  put  w  =  1,  J  =  |  in  the  above  equation  for  x ;  it  then 

becomes 

0-a^  +  ^'X^  +  ^a^  +  Q-x-^  =  % 

of  which  the  four  roots  are 

00,    oo,    +fV2,    -|V2; 

hence  the  approximate  values  of  the  finite  roots  are  ±  1.06. 

The  manner  in  which  the  infinite  branches  approach  their 
asymptotes  is  best  shoAvn  by  the  method  of  expansion,  in 
which  y  is  expressed  in  a  series  of  descending  powers  of  x. 

Write  the  equation  in  the  form 


^2  =  3a;-fV9^T^ 

=     -+     1     +  "  —    T  .    .  J 

X  2x^ 


166.] 


CURVE  TRACING 


287 


Hence 


This  verifies  the  equations  of  the  asymptotes  already 
found  ;  and,  moreover,  the  sign  of  the  third  term  shows  that 
the  curve   is  above  the  first  asymptote  for  large  positive 


Fio.  75. 


values  of  x,  and  below  it  for  large  negative  values.     On  the 
other  hand,  the  curve  is  below  the  second  asymptote  for 


288  DIFFERENTIAL   CALCULUS  [Ch.  XVIII. 

large  positive  values  of  x,  and  above  it  for  large  negative 
values. 

The  form  of  the  second  derivative  —^  is  too  complicated 

to  be  of  practical  use  in  determining  the  direction  of  bending. 

Since  each  infinite  branch  is  convex  to  its  asymptote  for 
large  values  of  a;,  hence  on  the  upper  right  hand  branch  the 
concavity  is  ultimately  upwards.  Near  the  origin  the  con- 
cavity is  downwards,  hence  there  must  be  a  point  of  inflexion 
on  this  branch,  and  also  on  the  branch  symmetrical  to  it. 

On  the  left  hand  branches  there  are  no  points  of  inflexion, 
for  if  there  were  one  on  either  branch  there  would  be  two 
on  that  branch,  and  it  would  then  be  possible  to  draw  a  line 
cutting  the  given  fourth  degree  curve  in  more  than  four 
points. 

167.  Form  of  a  curve  near  the  origin.  In  the  above  ex- 
ample, in  tlie  vicinity  of  the  origin,  the  curve  approaches 
the  form  of  an  ordinary  parabola  on  one  side  of  the  ^-axis 
(which  is  the  tangent  at  its  vertex),  and  has  a  cusp  of  the 
first  kind  on  the  other  side,  the  axis  of  x  being  the  cuspidal 
tangent. 

In  the  first  case  x  was  neglected  in  comparison  with  y, 

since  ^^"q-=  0;  while  in  the  second  case  i/  is  neglected  in 
comparison  with  a;,  since  „  ^  q^  =  0. 

In  many  cases  it  is  not  so  obvious  which  terms  can  be 
rejected,  especially  when  the  lowest  terms  in  the  expression 
are  of  high  degree. 

Before  proceeding  to  the  more  difficult  curves,  a  few  ele- 
mentary type  forms  will  be  given.  The  branches  of  every 
algebraic  curve  approximate  to  combinations  of  these  forms 
in  the  vicinity  of  any  assigned  point  as  origin. 


166-167.] 


CURVE  TRACING 


289 


1.    Trace  the  curve  y^  =  z. 
Here 


dx         2  x^' 


hence  the  slope  is  intinite  at  the 
origin,  and  diminishes  to  zero  at  in- 
finity, showing  that  the  curve  becomes  more  and  more  hori- 
zontal ;  the  bending  is  negative  on  the  upper  branch,  and 
positive  on  the  lower. 


Fig.  76. 


2.    The  curve  y  =  x^. 
Here 


dx 


<^;/_ 


dx"^ 


=  2, 


Fig.  77. 


hence  the  slope  is  zero  at  the  origin,  and 
becomes  infinite  at  infinity,  showing  that 
the  curve  becomes  more  and  more  vertical ; 
the  bending  is  always  positive. 

3.    The  curve  y  =.^. 

In  this  case    -^  =  Sa^, 
dx 

^-6x 

hence  the  slope  is  zero  at  the  origin, 

is  elsewhere    always   positive,   and 

becomes   infinite   at   infinity.      The   bending   changes   sign 

where  the  curve  passes  through  the  origin,  the  a;-axis  being 

the  inflexional  tangent. 


Fig.  78. 


290  DIFFERENTIAL   CALCULUS  [Ch.  XVIIL 

4.    Show  the  form  of  the  curve  y^  =  a?. 

Here         y  =  ±A  |  =  ±|^*.  ^^^^I^"*- 

The  curve  is  symmetrical  with  regard  to  the  x-axis ;  ana 
since  the  slope  is  zero  at  the  origin,  the  axis  of  x  is  tangent 
to  the  upper  and  the  lower  branch.  Since 
a  negative  value  of  x  makes  y  imaginary, 
the  curve  does  not  extend  to  the  left 
of  the  origin,  hence  there  is  a  cusp  of 
the  first  kind  at  this  point.  The  slope 
^^^'  '^'  increases  numerically  to  infinity  when  x 

becomes  infinite,  and  the  bending  is  always  positive  on  the 
upper  branch,  and  negative  on  the  lower.  This  curve  is 
called  the  semicubical  parabola  because  the  ordinate  is  pro- 
portional to  the  square  root  of  the  cube  of  the  abscissa. 

In  each  of  these  fundamental  types,  if  the  sign  of  either 
member  of  the  equation  be  changed,  the  curve  is  simply 
turned  over,  and  if  x  and  y  be  interchanged,  the  curve  is 
revolved  through  90  degrees. 

Some  more  complicated  cases  will  now  be  taken  up.  The 
following  general  principles  will  be  of  use. 

I.  When  the  equation  of  an  algebraic  curve  is  rationalized 
and  cleared  of  fractions,  if  the  constant  term  be  absent  the 
origin  is  on  the  curve ;  and  the  terms  of  the  first  degree, 
equated  to  zero,  give  the  equation  of  the  tangent  at  the 
origin. 

II.  If  the  constant  term  and  terms  of  the  first  degree  be 
absent,  the  origin  is  a  double  point ;  and  the  terms  of  the 
second  degree,  equated  to  zero,  give  the  equation  of  the  pair 
of  nodal  tangents. 


167.]  CURVE  TRACING  291 

III.  If  all  the  terms  below  the  third  degree  be  absent, 
the  origin  is  a  triple  point ;  and  the  terms  of  the  third 
degree,  equated  to  zero,  furnish  the  equation  of  the  three 
tangents  at  the  multiple  point.     Similarly,  in  general. 

For,  let  the  equation  be  of  the  form 

f(x,  y^=  ax  +  hy  ^- (cx^  +  dxy  +  ey"^)  +  ...  =  0,      (1) 

then  the  tangent  at  the  origin  will  be  represented  by  the 
equation 

dv 
in  which  ~  is  to  be  obtained  from  the  relation 
ax 

^  +  ^.$^  =  0;  (3) 

dx      by    ax 

dv 
hence,  eliminating  -~  between  the  last  two  equations,  the 

equation  of  the  tangent  at  the  origin  becomes 

dx      ^  dy         *  ^  ^ 

but  at  the  point  (0,  0), 

hence  the  equation  of  the  tangent  at  this  point  is 

ax  +  hy  =  Q.  (6) 

Again,  if  the  constants  a,  h  be  zero,  the  expression  for 
-^,  given  by  (.3),  is  indeterminate,  and  the  slope  at  the  origin 
is  to  be  obtained  from  the  quadratic  (Art.  161), 

dy^\dxj  '^     dxdy  \dxj  '^  dx^         '  ^  ^ 

DIFF.   CALC.  20 


292  DIFFERENTIAL    CALCULUS  [Ch.  XVIII. 

hence  eliminating  -^  between  (2)  and  (7),  there  results 
ax 

which  is  then  the  equation  of  the  pair  of  tangents  at  the 
origin  ;  but  at  the  point  (0,  0), 

^=2c    -^—=.d   ^=2 
b:^  ''   dxdy         '  dy^  ' 

hence  the  equation  of  the  pair  of  tangents  at  the  origin  is 

cx^ -{■  d  xy -^  ey^  =  Q  (9) 

Similarly  proceed  in  general. 

168.  Another  proof.  The  equation  that  gives  the  abscissas 
of  the  intersections  of  the  line  y  =  mx  with  the  given  curve  is 

(a  +  bm)x  +  ((?  +  dm  +  enfi^x^  +  •••  =  0. 

There  will  be  two  intersections  at  the  origin  if  a  +  5w  =  0, 

that  is,  if   m  —  —  -•     Hence  the  tangent  at   the  origin  is 
a 

Again,  if  a  =  0,  6  =  0,  every  line  through  the  origin  will 
meet  the  curve  in  two  coincident  points ;  and  in  this  case 
the  origin  will  be  a  double  point.  If,  moreover,  m  be  so 
taken  that  c  +  dm  +  errfi  =  0,  the  line  y  =  mx  will  meet  the 
curve  in  three  coincident  points  at  the  origin ;  hence  the 
equation  of  the  pair  of  nodal  tangents  is  to  be  found  by 
eliminating  m  between  y  =  mx  and  c  +  dm  +  em"^  =  0,  and 
is  therefore  cx^  +  dxy  +  ey^  =  0  ;  and  so  on. 

169.  Illustration.  Oblique  branch  through  origin.  Expan- 
sion of  y  in  ascending  powers  of  sc.     Given  the  equation 

y-2x  +  Sx^  +  4xy-  5y^-{-6s^  +  f=  0, 


167-169.]  CURVE  TRACING  293 

to  expand  3/  in  ascending  powers  of  x,  and  thence  to  trace 
the  locus  in  the  vicinity  of  the  origin. 

The  first  approximation  to  the  value  of  ^,  obtained  by 
omitting  terms  of  order  a^,  is  t/  =  2x,  which  is  the  equation 
of  the  tangent,  and  gives  the  direction  of  the  curve  at  the 
origin.  In  approaching  the  origin  along  the  curve,  the 
variables  x  and  y  are  infinitesimals  of  the  same  order,  and 

their  ratio  ^  =  2. 

X 

To  obtain  the  second  approximation  to  tlie  value  of  y,  test 
for  the  order  and  value  of  the  infinitesimal  y  —  2x,  by  com- 
paring it  with  x^^  thus 

=  9  when  x=  0,  y  =  0,  ^=2, 


w  —   *> 


hence  1/  =  2x  +  ds^y  with  an  error  above  the  second  order 
of  smallness. 

Since  the  second-order  term  9a:2  {§  positive,  the  curve  is 
situated  above  the  tangent  y  =  2x  on  both  sides  of  the 
origin.  If  desired,  the  third-order  term  can  be  obtained  by 
substituting  2  3^  -f-  9  a:^  for  y  in  the  second  and  third  order 
terms  of  the  given  equation,  and  collecting  the  coefficient  of 
2^.  The  third  approximation  is  then  y  =  2a;-f-  9a:2  +  130 2:^. 
This  shows  that  the  curve  is  above  the  parabola  y  —  2x-\-^:i^ 
on  the  right  of  the  origin,  and  below  it  on  the  left.  These 
two  curves  have  contact  of  the  second  order  at  the  origin. 

(2)  Trace  in  the  vicinity  of  the  origin  the  curve 

-27?  ^xy  -\-y'^-\-:fi-2f  ^3^-2^y  =  ^. 

Here  the  origin  is  a  double  point,  at  which  the  tangents, 
obtained  by  factoring  —  2  a^  +  a;y  -|-  y^  =  0,  are  ^  —  a;  =  0, 


294 


DIFFERENTIAL   CALCULUS 


[Ch.  XVIII. 


y  +  2x  =  0  ',  hence  on  one  branch  ^  =  1,  and  on  the  other 

X 

^  =  —  2  ;  thus  the  branches  are  oblique,  and  on  each  branch 

X 

X  and  y  are  infinitesimals  of  the  same  order. 

The  second  approximation  to  the  equation  of  each  branch 
is  to  be  obtained  by  taking  account  of  the  third-order  terms 
in  thie  given  equation,  thus 

(t/-a.)(^  +  2x)  =  - 0^5  +  23/3; 

then,  on  the  first  branch  the  comparison  oi  y  —  x  with  a^  gives 

v3 


y  -^ 


-1  +  2 


2  + 


y 


1 

3' 


e- 


hence  the  branch  has  the  approximate  equation  y  =  x  +  ^aP^ 
which  shows  that  it  lies  above  the  tangent  y  =  x  on  both 
sides  of  the  origin. 

The  third  approximation,  obtained  by  writing  the  given 

equation  in  the  form 

_     _—a^-\-2y^  —  x!^  +  2.a?y 
y~^'^  y  -\-'lx  "' 

substituting  for  y  the  second  approximation  and  dividing 
as  far  as  a^,  is  y  =  a;  +  |^  a^  +  |  f  a^ ; 
which  shows  that  the  first  branch  is 
above  the  parabola  y  —  x  -\-\x^  on 
the  right,  and  below  it  on  the  left 
of  the  origin. 

On   the    second    branch    the   com- 
parison of  y  -\-1x  with  :i^  gives 


Fig.  80. 


y+2a; 


-1  +  2 


y-\ 

X 


17 
3' 


B- 


1G9-170.] 


CURVE  TRACING 


295 


hence  its  approximate  equation  is  2/  =  — 2x  +  ^3^. 
third  approximation  isi/  =  ~2x-\-^x^  —  ^^^-  7?. 
Both  branches  are  shown  in  Fisr.  80. 


The 


170.   Branches  touching  either  axis.     Trace,  near  the  origin, 

tlie  curve 

x'lf'  +  7?y  —  «/"*  —  2  2^  =  0. 

The  i/-axis  is  a  single  tangent,  and  the  a:-axis  is  a  double 
one  ;  thus  the  origin  is  a  triple  point. 

To  determine  the  form  of  the  curve  near  the  origin,  the 
method  of  Art.  169  will  not  apply,  as 
x^  y  are  not  infinitesimals  of  the  same 
order  on  either  branch.    Here  a  method 

of  trial  will  be  employed.     Suppose  the 

terms  xy^  and  y^  are  of  the  same  order 
on  one  branch,  then  x  and  y^  are  of  the 
same  order,  i.e.,  y  is  of  the  same  order 
as  a;^  hence  the  terms  in  the  given 
equation  are  of  the  respective  orders 

2,31,2,5; 

thus  the  terms  selected  are  of  the  lowest  order,  and  are  there- 
fore the  controlling  ones  near  the  origin,  showing  that  there 
is  a  branch  having  the  approximate  equation  xy^  —  y^  =  0. 
Removing  the  factor  y^,  the  equation  of  this  branch  is 
X  —  y^  =  0 ;  and  the  next  term  in  its  equation  is  given  by 


V 


Fio.  81. 


2         ^^ 


^y  ^ 


y 


f  + 


hence  the  branch  is  situated  to  the  left  of  the  parabola 
X  —  y^  =  Q  above  the  a;-axis,  and  to  the  right  below  that 
axis. 


296 


DIFFERENTIAL   CALCULUS 


[Ch.  XVIII. 


Next  suppose  there  is  a  branch  for  which  y^  and  2  a^  are 
infinitesimals  of  the  same  order ;  then  y  has  the  same  order 
as  a;*,  and  the  four  terms  have  the  orders 

31,       4^,       5,       5; 

hence  there  is  no  branch  for  which  the  two  terms  selected 
are  the  controlling  ones. 

Once  more,  suppose  there  is  a  branch  on  which  xy^  and 
Qi^y  are  of  the  same  order;  then  dividing  by  xy,  it  follows 
that  y  is  of  the  same  order  as  x^,  and  the  orders  in  x  of 
the  four  terms  are 

O,        o,        o,        o. 

Therefore  there  is  a  branch  on  which  the  first,  second,  and 
fourth  are  the  controlling  terms,  and  its  approximate  equa- 
tion is 

xy^  +  a^y  —  2  a;^  =  0, 

which  reduces  to 


I.e. 


y2  +  a;2y  -  2  a;4  =  0, 


Fig.  82. 


y-  ^  = 


x(^y  -\r  2a;2^ 


Hence  the  part  of  the  curve  in 
question  consists  of  the  two  parab- 
olas y  =  x^,  y  =  —  ^x^. 

Writing  the  given  equation  in 
the  form 

xiy-^')(.y  +  ^ofi)=y\ 

the  first  of  these  branches  has  the 
equation 


=  (approx.) 


3? 
x(j^+2x^) 


i^-, 


170-171.] 


CURVE  TRACING 


297 


hence  the  curve  gets  steeper  than  the 
approximate  parabola  on  one  side  and 
flatter  on  the  other.  Similarly,  the 
approximate  equation  of  the  other 
branch  is 

Combining  these  two  sets  of  results, 
the  form  of  the  curve  in  the  vicinity 
of  the  origin  is  as  given  in  Fig.  83. 


171.    Two    branches    oblique ;     a    third  touching   jr-axis. 

Trace,  in  the  vicinity  of  the  origin,  the  curve 

3^  +  7?y  —  ^  =  0. 

Since  there  are  no  terms  below  the  third  degree,  the  origin 
is  a  triple  point,  and  the  three  tangents  represented  by  the 
equation 

Q^y  -  f  =  y (x  -  y^{x  +  y^  =  0 

have  the  separate  equations 

y  =  {)^     y  =  X,    y  -  -  X. 

To  show  roughly,  without  resorting  to  expansion,  how  the 
curve  is  related  to  these  three  lines,  write  its  equation  in  the 
form 

y(j/^  -  2^2)  =  a:*. 

First  consider  points  near  the  origin  on  the  branch  that 
touches  the  line  ^  =  0.  Here  3.  J"q^=  0,  hence  y  is  infini- 
tesimal as  to  a;,  and  the  factor  (^  —  ar^)  is  negative,  but  the 
term  on  the  right  is  positive,  hence  the  other  factor  on  the 
left,  3/,  is  negative ;  thus  the  curve  is  below  the  line  y=0 
on  both  sides  of  the  ^-axis. 


298 


DIFFERENTIAL   CALCULUS 


[Ch.  XVIII. 


Next  consider  points  on  the  branch  that  touches  the  line 
y  ■=  X.  When  x  is  positive,  y  is  positive,  hence  the  factor 
y"^  —  x^  is  also  positive ;  thus  y  is  greater  than  a;,  and  the 
curve  lies  above  the  tangent  y  =  x  in  the  first  quarter.  In 
the  third  quarter  both  x  and  y  are  negative,  hence  y^  —  x^  \s 
negative,  and  y  is  numerically  less  than  x\  thus  the  curve  is 
above  the  tangent  y  =  x'ln  the  third  quarter. 

Lastly,  consider  points  on  the  branch  that  touches  the  line 
y  =  —  x.  Here  again  (jy^  —  a;^)  has  the  same  sign  as  y^  hence 
in  the  second  quarter  y  is  numerically  greater  than  x,  and 
the  curve  is  above  the  tangent ;  but  in  the  fourth  quarter  y 
is  numerically  less  than  x,  and  the  curve  is  again  above  the 
tangent. 

The  position  of  the  three  branches  can,  however,  be  ascer- 
tained with  greater  accuracy  from  their  approximate  equa- 
tions, obtained  by  the  method  of  expansion :  y  =  —  x^  —  x^  •••; 

y=x  +  ^2P-^a^+'-;  y  =  -x  +  ^a^  +  ^a^"'. 
The  form  of  the  infinite  branches  will  be  considered  later, 
and  it  will  appear  that  the  branches  in  the  first  and  second 
quarters  are  the  only  ones  that  ex- 
tend to  infinity  (Fig.  84). 

Form  of  curve  in  vicinity  of  point 
(a,  6).  The  form  in  the  vicinity  of 
any  point  («,  6)  can  be  found  by 
first  transforming  to  (a,  5)  as  origin, 
and  proceeding  as  in  Arts.  169-171. 
This  is  equivalent  to  expanding  the 
given  function  f(x,  y)  in  powers  of 
x  —  a^  y  —h\  and  then  expressing 
h  in  ascending  powers  of  the  small 


Fig.  84. 


the  small  number  y 
number  x  —  a. 

Remark  on  expansion  of  implicit  functions. 


The  methods 


171-172.]  CURVE  TRACING  299 

of  Arts.  139-171  are  often  practically  useful  in  purely  alge- 
braic operations.  They  may  evidently  be  applied  to  any 
implicit  relation  between  x  and  y,  when  the  object  is  to  ex- 
press either  variable  explicitly  in  terms  of  the  other,  in  the 
vicinity  of  two  given  corresponding  values  (aj  =  a,  y  =  6), 
to  any  required  degree  of  approximation. 

EXERCISES 

Examine  the  following  curves  in  the  vicinity  of  the  origin  and  find 
two  or  three  terms  of  the  expansion  of  y  in  ascending  powers  of  x : 

1.  y2  =  a;<  +  x6;  3.   y'i  =  2x^  +  x»;         5.   y^  -  x^  =  x^ -2xhf  -  3*\ 

2.  y^  =  2x^y  +  x^;        ^.   y«  =  x*  -  x*;  6.   9xy  =  23^  +  2y^ 

7.  In  No.  6,  for  the  vicinity  of  the  point  (1,  2),  expand  y  —  2  in  as- 
cending powers  of  a;  —  1. 

172.   Approximation  to  form  of  infinite  branches.     It  has 

been  shown  that  in  the  vicinity  of  the  origin,  the  approxi- 
mate form  of  each  branch  of  the  curve  could  be  obtained 
by  examining  the  different  suppositions  regarding  the  rela- 
tive orders  of  the  infinitesimals  x  and  y,  in  consequence  of 
which  two  or  more  terms  of  the  equation  should  become 
infinitesimals  of  like  order,  and  compared  with  these  all  the 
other  terms  should  be  of  higher  order,  and  could  therefore 
be  neglected  in  writing  down  the  first  approximation  to  the 
value  of  y  in  ascending  powers  of  x. 

On  a  similar  principle  the  approximate  form  of  each 
branch  of  the  curve  at  great  distances  from  the  origin  can 
be  obtained  by  examining  the  different  suppositions,  regard- 
ing the  relative  orders  of  the  infinites  x  and  y,  in  conse- 
quence of  which  two  or  more  terras  should  become  infinites 
of  the  same  order,  and  in  comparison  with  these  all  the  other 
terms  should  be  infinites  of  lower  order,  and  could  therefore 


300 


DIFFERENTIAL   CALCULUS 


[Ch.  XVIII. 


be  neglected  in  writing  down  the  first  approximation  to  the 
value  of  y  in  descending  powers  of  x. 

Ex.  1.    Take  the  curve  traced  near  the  origin  in  Art.  171, 

y(^x^  —  y'^)  +  X*  =  0. 

Here  the  supposition  that  y  is  an  infinite  of  the  same  order  as  x* 
makes  the  terms  y^  and  x*  infinites  of  order 
4,  and  the  term  yx^  an  infinite  of  order  3^; 
thus  there  is  an  infinite  branch  which  has 
the  approximate  equation  y^  =  x*,  and  hence 
passes  out  of  the  field  in  the  manner  shown 
in  Fig.  85. 

Again,  the  supposition  that  y  is  of  the 
same  order  as  x'^  makes  yx^  and  x*  infitdtes 
of  order  4,  and  the  term  y^  an  infinite  of 
order  6,  which  cannot  be  neglected  in  com- 
pai'ison.  Hence  there  is  no  infinite  branch  on 
which  y  is  a^iproximately  proportional  to  x^. 

Similarly  the  third  supposition  does  not  correspond  to  an  infinite 
branch. 

Ex.  2.  Consider  the  curve  that  was  traced 
near  the  origin  in  Art.  170, 

xy^  +  x^y  -  y*  -2x^  =  0. 

The  supposition  that  y  is  of  the  same 
order  as  x^  makes  the  four  tei'ms  of  the 
orders  3 J,  4 J,  5,  5;  hence  there  is  an  infi- 
nite branch  whose  approximate  equation  is 
y*  +  2  x^  =  0.     The  form  of  the  curve  is  shown  in  Fig 


Fio.  85. 


Fio.  86. 


Hyperbolic  and  parabolic  branches.  Expansion  in  descend- 
ing series.  On  an  infinite  branch  the  coordinates  x  and  y 
may  behave  as  follows  : 

1.  One  of  the  coordinates  may  approach  a  finite  number, 
and  the  other  become  infinite.  The  branch  has  then  a  hori- 
zontal or  vertical  asymptote  (Art.  130),  and  is  thus  a  hori- 
zontal or  vertical  hyperbolic  branch  (Fig.  40). 


172.]  CURVE  TRACING  301 

2.  The   coordinates   may    become   infinites   of  the   same 

y  •  • 

order.      Then  -  =  w,  a  finite  number ;    hence  there  is,  in 

general,  an  oblique  asymptote,  that  is,  the  infinite  branch 
is,  in  general,  an  oblique  hyperbolic  branch.  [In  a  special 
case  it  is  an  oblique  parabolic  branch.     See  Exs.  4,  5.] 

3.  The  coordinates  may  become  infinites  of  different  or- 
ders. If  y  is  an  infinite  of  higher  order  than  a;,  there  is  a 
parabolic  branch  on  which  the  tangent  tends  to  become  ver- 
tical (Figs.  85,  86),  —  called  a  vertical  parabolic  branch. 
If  y  is  of  lower  order  than  a;,  there  is  a  horizontal  parabolic 
branch  (Fig.  81). 

The  test  for  Case  (1)  has  been  given  in  Art.  131.  Case 
(2)  comes  under  the  head  of  oblique  asymptotes  ;  but  it  may 
be  conveniently  treated  along  with  Case  (3)  by  the  method 
of  this  Article.  The  test  for  Case  (2)  is  to  observe  whether 
there  are  two  or  more  terms  of  the  highest  degree  in  x 
and  y.  If  so,  the  supposition  that  y  is  of  the  same  order  as 
X  makes  these  the  controlling  terms. 

Ex.  3.  Test  for  oblique  infinite  branches  the  fourth-degree  curve 
a;*  +  a:^  +  2  ^8  =  x8  +  3  a;2  -  3/2. 

Here  there  are  two  fourth-degree  terms,  and  the  supposition  that  y 
and  X  are  infinites  of  the  first  order  makes  these  the  controlling  terms; 
hence  there  is  an  oblique  branch  on  which  "  =  —  1.     On  putting  the  first 

X 

approximation,  ^=  —  ar,  in  the  third-degree  terms,  and  dividing  by  ar*,  there 
results,  for  the  second  approximation,  w  =  —  x  -|-  3 ;  and  this,  when  used 

1  R 

in  the  same  way,  gives  the  third  approximation,  y  =  —  x-f3 1-  •••. 

X 

Thus  the  branch  is  hyperbolic,  having  the  oblique  asymptote  y=— z-f3. 
There  is  also  a  pair  of  vertical  parabolic  branches,  on  which 


302  DIFFERENTIAL   CALCULUS  [Ch.  XVIII. 

Ex.  4.   Test  in  the  same  way  the  cubic  curve 

(y  -2xy^(i/  +  x)  =  dx"^  +  xy  +  5ij^  +  3x  -7  y  +  8, 

in  which  the  terms  of  the  third  degree  have  a  square  factor. 

Corresponding  to  the  single  factor  y  +  x  there  is,  as  before,  a  hyper- 

7 

bolic  branch  whose  equation  is  y  =  —  x  +  1 +  ••'. 

9  X 
The  equation  of  the  branch  corresponding  to  the  square  factor  is 
given  by 

(^      ^.,^5x^  +  xy  +  5y^+Sx-7y  +  8 
^^      ~    ^  y  +  x 

The  first  approximation,  y  =  2  x,  used  on  the  right,  gives  (y  —  2  x)^ 
=  9  x;  and  the  second  approximation,  y  =  2  x  ±3xi,  used  in  the  same 
way,  gives  y  =  2x  ±  3  x^  +  2  +  •••  in  descending  powers  of  xi.  Hence  the 
branch  on  which  "  =  2  has  no  linear  asymptote.     The  curvilinear  asynip- 

X 

tote  of  lowest  degree  is  the  second-degree  curve  {y  —  2  x  —  2)^  =  9  x. 
There  are  thus  two  oblique  parabolic  branches. 

Ex.  5.  When  the  terms  of  highest  degree  have  a  factor  repeated  three 
times,  show  that  the  corresponding  expansion  of  y  descends  in  powers 
of  X3,  and  that  the  asymptote  of  lowest  degree  is  a  cubic  curve. 

The  method  of  successive  approximation  in  descending 
series  can  also  be  used  in  Case  (3),  when  once  the  first  ap- 
proximation has  been  obtained  by  the  method  of  comparison 
given  above. 

Ex.  6.  In  the  curve  of  Fig.  85,  the  first  approximation  is  y  =  x*. 
Substituting  this  on  the  right  of  y^  =  x*  +  x'^y,  and  taking  cube  root,  the 
second  approximation  \s  y  —  x^  -\-  \ x^  +  •••,  in  descending  powers  of  ars. 

4  2 

For  the  third  term  it  is  easiest  to  let  y  —  x'^  +  \x^  +  p,  substitute  in 
y^  =  X*  +  x^y,  and  determine  p  so  that  the  coefficients  shall  be  equal  as 
far  down  as  x^;  then  />  =  —  ^y ;  and  y  =  x^  -\-  ^x*  —  ^j  +  •••. 

Ex.  7.  In  Ex.  5  of  Art.  171  show  that  on  two  branches  the  controlling 
terms  are  y^  +  2  x^y  -  x*;  that  is,  [y  -  x^  (V2  -  1)]  [y  -f-  x^(V2  +  1)], 
and  that  the  equations  of  these  branches  are 

y=(±V2-l)x2T-^.±-^r+-.. 
2V2     16  V2 


172-173.]  CURVE  TRACING  303 

Remark  on  implicit  functions.  By  this  method,  when  any 
implicit  algebraic  relation  between  x  and  y  is  given,  the 
value  of  either  variable  for  large  ^values  of  the  other  can  be 
computed  by  descending  series,  with  small  relative  error. 

Transcendental  Cartesian  Curves.  A  number  of  figures 
of  important  transcendental  curves  are  shown  on  pp.  237-238, 
and  in  A.  G.,  p.  211  ff.     They  are  traced  by  tabulating  y, 

with  assistance  from  ^,  —^■^ 
ax    dor 

EXERCISES 

Apply  the  methods  of  this  article  to  the  equations  at  end  of  Art.  171. 
In  No.  6  compute  the  value  of  y  when  x  =  20,  by  descending  series. 

173.  Curve  tracing :  polar  coordinates.  In  tracing  curves 
defined  by  polar  equations  there  is,  as  in  the  case  of  Cartesian 
equations,  no  fixed  method  of  procedure. 

If,  as  usually  happens,  the  equation  can  be  solved  for  p, 
successive  values  may  be  given  to  ^,  and  the  corresponding 
values  of  p  computed  and  tabulated.  In  constructing  the 
table  it  is  useful  to  record  at  what  values  of  d  the  radius 
vector  p  has  turning  values.  The  critical  values  of  6  for 
this  purpose  are,  as  usual,  determined  from  the  equations 

-^  =  0,    -^  =  oo  ;    and  are   separately  tested   by   observing 

whether  the  derivative  changes  its  sign. 

Next  should  be  noted  the  asymptotic  directions,  which 
correspond  to  those  values  of  0,  if  any,  at  which  p  passes 
through  an  infinite  value.  The  distance  of  the  asymptote 
from  the  infinite  radius  vector  is  given  in  magnitude  and 
sign  by  the  corresponding  value  of  the  polar  subtangent 

<T  =  p^  — -.     Again,  if  p  tends  to  a  definite  limit,  as  6  becomes 
dp 

infinite,  there  is  a  circular  asymptote. 


304 


DIFFERENTIAL   CALCULUS 


[Ch.  XVIIL 


On  sketching  the  path  of  the  point  (p,  ^)  from  the  tabu- 
lated record,  greater  accuracy  in  the  direction  of  the  curve 
at  any  point  may  be  obtained  by  computing  the  slope  of 
the  tangent  line  to  the  radial  direction,  from  the  relation 

tani/r  =  /3— .     The  same  result  can  be  achieved  by  tabulating 

the  values  of  <t^  the  polar  subtangent,  not  merely  for  asymp- 
totic directions,  but  for  other  convenient  values  of  6. 
Assistance  in  tracing  the  curve  may  sometimes  be  obtained 
by  noticing  whether  there  are  any  axes  of  symmetry. 


Ex.  1.   Trace  the  locus  of  the  equation 


Here 


p  =  a(sec  26  +  tan20)  =  a 


1  +  sin  2  g 
cos  2  ^ 


^  =  2asec2^(tan2d  +  sec2^)=2al-±4^' 
dO  ^  ^  cos22^ 

de 


tan  \}i  =  p  —  =  i  cos  2  6, 
dp 

<T  =  p  tan  1/^=2  «(1  +  sill  2  6), 

whence  the  following  table  may  be  constructed,  and  the  locus  traced  ; 


0 

P 

dp 

^  de 

tan  1^ 

<r 

0 

a 

2a 

.5 

.5  a 

iTT 

3.7  a 

14.8  a 

.25 

Ma 

iTT 

1^ 

> 

> 

a 

iTT 

-  3.7  a 

14.8  a 

-  .25 

Ma 

^T 

—  a 

.   2a 

-.5 

.5  a 

TT 

~0 

+ 

a 

a 
2n 

"0 

+ 

.5 

^0 

+ 

.5  a 

Irr 

3.7  « 

14.8  a 

.25 

.93  a 

Itt 

+ 

t« 

■^0 

a 

asj'raptote 


asymptote 


1730 


CURVE  TRACING 


305 


As  $■  increases  from  0  to  J  tt,  the  tracing  point  P  moves  from  A  to  B, 
and  as  6  increases  by  successive  steps  to  Jtt,  Itt,  tt,  ^tt,  f  tt,  ^tt,  27r,  P 


Fifi.  8T. 


moves  respectively  from  B'  to  C,  C  to  D,  D  to  E,  E  to  F,  P  to  G,  G  to 
D,  D  to  A.     The  lines  6  =  ^Tr,  d=^Tr  are  axes  of  symmetry. 

Ex.  2.   Transform  to  polar  coordinates  the  equation 

(x2  +  y2)2  -  2  ay(x^  +  y^)  =  a^'^, 

and  then  trace  the  curve. 

On  putting  x  =  p  cos  6,  y  =  /o  sin  6,  dividing  by  p\  and  solving  the 
quadratic  for  p,  there  results 

p  =  a(sin  $  ±  1). 

First  take  the  upper  sign  ;  then 

^  =  acosO,   tan,A=fi^  =  ?illi±i, 
du  dp  cos  d 

and  the  following  table  is  easily  computed. 

The  figure  is  shown  in  Art.  108.  If  the  lower  sign  be  taken,  the  same 
curve  will  be  traced  in  a  different  order.  The  line  ^  =  |  ir  is  an  axis  of 
symmetry. 


306 


DIFFERENTIAL   CALCULUS        [Ch.  XVIII.  173. 


e 

p 

de 

tan  iji 

0 

a 

a 

1 

iT 

1.7  a 

.11  a 

2.41 

^^ 

1.87  a 

.5  a 

3.73 

l^ 

2a 

-"o       , 

+ 
CO 

fTT 

1.7  a 

-.71a 

-2.41 

TT 

a 

—  a 

-1 

fTT 

.29  « 

-.71a 

-.41 

.29  a 

"0 
.71a 

"0 

+ 

.41 

Y^ 

.5  a 

.87  a 

.58 

27r 

a 

a 

1 

p  a  maximuin,   i/^  =  |  ir. 


p  a  minimum,   i/'  =  0, 
origin  a  cusp. 


EXERCISES 


Trace  the  following  curves : 

X 


1.  y 
y 


1  +X2 

2.   .?/2  =  2  a;2  +  a;3, 

1/2 


4.   3/2(x  _  fl)  =  (x  +  o)  x^. 
3.   2/2  =  a:^  +  x^.  5     a:2^2  ^  a2(-^2  _  ^2), 

6.  Show  that  the  curve  y^  =  x^  —  x*  has  two  branches  which  are 
both  tangent  to  the  axis  of  x  at  the  origin. 

7.  Determine  the  direction  of  the  curve  y^  =  x'^(^x  —  a)  at  each  point 
where  it  crosses  the  axis  of  x. 

8.  Trace  the  curve  y^  —  axy  —  b^x  =  0  in  the  neighborhood  of  the 
origin. 

9.  Show  that  the  curve  p  =  1  +  sin  5  6  consists  of  5  equal  loops. 

10.  Trace  the  curve  p  cos  2  ^  =  a. 

Find  its  asymptotes  and  lines  of  symmetry. 

11.  Trace  the  curve  p  =  a  (tan  ^  —  1 ) . 

■1 =  C*-3. 


1      V-  1        - 
12.    Trace  the  curves  y  =  e'',     y  =  e*, =  e', 


y 


y 


13.  Find  the  points  of  inflexion  of  the  curve  y  =  e~^'^. 
This  curve  is  known  as  the  probability  curve  (Fig.  49). 

14.  p=  a  +  sin  f  0.  15.   p  =  a  (1  -  tan  &). 


CHAPTER   XIX 

ENVELOPES 

174.   Family  of  curves.     The  equation  of  a  curve, 

f(x.  y-)  =  0, 

usually  involves,  besides  the  variables  x  and  y,  certain  coeflfi- 
cients  that  serve  to  fix  the  size,  shape,  and  position  of  the 
curve.  The  coefficients  are  called  constants  with  reference 
to  the  variables  x  and  «/,  but  it  has  been  seen  in  previous 
chapters  that  they  may  take  different  values  in  different 
problems,  while  the  form  of  the  equation  is  preserved.  Let 
a  be  one  of  these  "constants"  ;  then  if  a  be  given  a  series 
of  numerical  values,  and  if  the  locus  of  the  equation  be 
traced,  corresponding  to  each  special  value  of  a,  a  series  of 
curves  is  obtained,  all  having  the  same  general  character, 
but  differing  somewhat  from  each  other  in  size,  shape,  or 
position.  A  system  of  curves  so  obtained  by  letting  one  of 
the  constant  letters  assume  different  numerical  values  in  the 
fixed  form  of  equation  /  (a;,  y)  =  0  is  called  a  family  of 
curves. 

Thus  if  A,  h  be  fixed,  and  jt)  be  arbitrary,  the  equation 
(jy  —  k)^=2 p(x—  1i)  represents  a  family  of  parabolas,  having 
the  same  vertex  (A,  A-),  and  the  same  axis  y  =  Tc,  but  having 
an  arbitrary  latus  rectum.  Again,  if  k  be  the  arbitrary 
constant,  this  equation  represents  a  family  of  parabolas 
having  parallel  axes,  the  same  latus  rectum,  and  having 
their  vertices  on  the  same  line  x  =  h. 

DIFF.  CA1.C.  —21  307 


308  DIFFERENTIAL   CALCULUS  [Cii.  XIX. 

The  presence  of  an  arbitrary  constant  a  in  the  equation  of 
a  curve  is  indicated  in  functional  notation  by  writing  the 
equation  in  the  form  f(x^y^  «)  =  0.  The  quantity  a,  which 
is  constant  for  the  same  curve  but  different  for  different 
curves,  is  called  the  parameter  of  the  family.  The  equations 
of  two  neighboring  members  are  then  written 

f(x,  y,  o)  =  0,   /(a-,  y,a  +  li)  =  0, 
in  which  A  is  a  small  increment  of  a  ;  and  these  consecutive 
curves  can  be  brought  as  near  to  coincidence  as  desired  by 
diminishing  h. 

175.  Envelope  of  a  family  of  curves.  The  locus  of  the 
points  of  ultimate  intersection  of  consecutive  curves  of  a 
family,  when  these  curves  approach  nearer  and  nearer  to 
coincidence,  is  called  the  envelope  of  the  family. 

Let  f(x,  y,  a)  =  0,         f(x,  y,a  +  h)=0  (1) 

be  two  curves  of  the  family.  By  the  theorem  of  mean 
value  (Art.  66) 

f(x,  y.  a  +  A)  =f(x,  y,  a)  +  A^(a:,  y,  a  +  OK),    (2)  [0  <  ^  <  1 

but  the  points  common  to  the  two  curves  satisfy  equations  (1), 

and  therefore  also  satisfy  -~-(x,  y,  a  +  61i)=  0.     Hence,  in 

the  limit,  when  A  =  0,  it  follows  that  ~  {x,  y,  a)  =  0  is  the 

equation  of  a  curve  passing  through  the  ultimate  intersec- 
tion of  the  curve  f(x,  ?/,«)=  0  with  its  consecutive  curve. 
This  determines  for  any  assigned  value  of  «  a  definite  point 
of  ultimate  intersection  on  the  corresponding  member  of  the 
family.  The  locus  of  all  such  points  is  then  to  be  obtained 
by  eliminating  the  parameter  a  between  the  equations 

f{x,  y,  «)  =  0,  ^(a:,  y,  «)=  0. 


174-1 7G.]  ENVELOPES  309 

The  resulting  equation  is  of  the  form  F(^x^  y')  =  0,  and 
represents  the  fixed  envelope  of  the  family. 

176.  The  envelope  touches  every  curve  of  the  family. 

0 


I.  Geometrical  proof.  Let  A,  B,  O  be  three  consecutive 
curves  of  the  family  ;  let  A,  B  intersect  in  P  ;  B,  0  inter- 
sect in  Q.  When  P,  Q  approach  coincidence,  PQ  will  be 
the  direction  of  the  tangent  to  the  envelope  at  P ;  but  since 
P,  Q  are  two  points  on  B  that  approach  coincidence,  hence 
P^  is  also  the  direction  of  the  tangent  to  P  atP ;  thus  B 
and  the  envelope  have  a  common  tangent  at  P ;  similarly 
for  every  curve  of  the  family. 

II.  More  rigorous  analytical  proof.     Let  ^/(a?,  y,  «)=  0 

be  solved  for  a,  in  the  form  «  =  <^(x^  y) ;  then  the  equation 
of  the  envelope  is 

/(«,  y,  <^(a;,  y))=0. 

Equating  the  total  a;-derivative  to  zero, 

dx      dy  dx      dipKdx      by  dx)        * 

hf      bf 
but  ~  =  ~-=:0^  hence  the  slope  of  the  tangent  to  the  en- 
velope at  the  point  (x,  y')  is  given  by 

^x      dy  dx        ' 

but  the  same  equation  defines  the  direction  of  the  tangent  to 
the  curve  /(x,  y^  «)=  0  at  the  same  point.      Therefore  a 


310  DIFFERENTIAL   CALCULUS  [Ch.  XIX. 

point  of  ultimate  intersection  on  any  member  of  the  family 
is  a  point  of  contact  of  this  curve  with  the  envelope. 

Ex.     Find  the  envelope  of  the  family  of  lines 


obtained  by  varying  m. 
Differentiate  (1)  as  to  m, 


y  =  mx  +  ^,  *    (1) 


0  =  ^-£-  (2) 


Hence  the  line  (1)  meets  its  consecutive  line  where  it  meets  (2).  To 
eliminate  m,  solve  (2)  for  m,  substitute  in  (1),  and  square;  then  the  locus 
of  the  ultimate  intersections  is  the  fixed  parabola 

y^  =  Apx. 

177.  Envelope  of  normals  of  a  given  curve.  The  e volute 
(Art.  158)  was  defined  as  the  locus  of  the  center  of  curva- 
ture. The  center  of  curvature  was  shown  to  be  the  jDoint  of 
intersection  of  consecutive  normals  (Art.  151),  hence  by 
Art.  175,  the  envelope  of  the  normals  is  the  evolute. 

Ex.   Find  the  envelope  of  the  normals  to  the  parabola  y^  =  Apx. 
The  equation  of  the  normal  at  (x,,  ?/j)  is 

y-yi  =  ^\^-^i)^ 

or,  eliminating  Xj  by  means  of  the  equation  j/j^  =  4  px^, 


The  envelope  of  this  line,  when  y^  takes  all  values,  is  required. 
Differentiating  as  to  y^, 

-1  = 

up'-     '^p 


_^yx' 


Substituting  this  value  for  y^  in  (1),  the  result, 
27py^  =  i(x-2py, 
is  the  equation  of  the  required  evolute. 


176-178.]  ENVELOPES  311 

178.  Two  parameters,  one  equation  of  condition.  In  many 
cases  a  family  of  curves  may  have  two  parameters  which  are 
connected  by  an  equation.  For  instance,  the  equation  of 
the  normal  to  a  given  curve  contains  two  parameters,  a:^,  y^, 
which  are  connected  by  the  equation  of  the  curve.  In  such 
cases  one  parameter  may  be  eliminated  by  means  of  the 
given  relation,  and  the  other  treated  as  before. 

When  the  elimination  is  difficult  to  perform,  both  equa- 
tions may  be  differentiated  as  to  one  parameter  a,  regarding 
the  other  parameter  j9  as  a  function  of  a,  giving  four  equa- 

dB 
tions  from  which  a,  ^,  and  -^  may  be  eliminated,  and  the 

da 
resulting  equation  will  be  that  of  the  desired  envelope. 

Ex.  1.   Find  the  envelope  of  the  line 

a     0 

the  sum  of  its  intercepts  remaining  constant. 
The  two  equations  are 

-  +  !=!' 
a     0 


Differentiate  as  to  a, 


a-\-b 


-X      y  db  _Q 

1+^  =  0; 
da 


db 


eliminate  — ,  then  ~==^,  therefore 
da  a^     b^ 


X     y     x^y 
a     b      a      b      \ 


a      b     a  +  b      c 
therefore  Vx  +  Vy  =  Vc 

is  the  equation  of  the  desired  envelope. 


hence     a  =  Vex,     b  =  "s/cy ; 


312 


DIFFERENTIAL   CALCULUS 


[Ch.  XIX. 


Ex.  2.   Find  the  envelope  of  the  family  of  coaxial  ellipses  having  a 
constant  area. 

Here  ^  +  !i=l5 

ah  =  k\ 

For  symmetry,  regard  a  and  b  as  functions  of  a  single  parameter  t, 
then 

^  da  +  -^  (lb  =  0, 


hence 


hda  +  adb  =  0 ; 
fjp.  ~  62  -  2' 


a  =  ±a:V^,     b=±y\/2, 
and  the  envelope  is  the  pair  of  rectangular  hyperbolas  xy  =  ±  |  jfc*. 


Fig.  89 


Note.  A  family  of  cxirves  with  a  single  parameter  may  have  no 
envelope ;  i.e.,  consecutive  curves  may  not  intersect ;  e.g.,  the  family  of 
concentric  circles  x^  -^  y"^  =.  r^,  obtained  by  giving  r  aU  possible  values. 


178.]  ENVELOPES  313 


EXERCISES 

1.  Find  the  envelope  of  the  parabolas  y'  =  —  (x  —  a),  a  being  a 
parameter.  ^ 

2.  A  straight  line  of  fixed  length  a  moves  with  its  extremities  in 
two  rectangular  axes ;  find  its  envelope, 

3.  Ellipses  are  described  with  common  centers  and  axes,  and  having 
the  sum  of  the  semi-axes  equal  to  c.     Find  their  envelope. 

4.  Find  the  envelope  of  the  straight  lines  having  the  product  of 
their  intercepts  on  the  coordinate  axes  equal  to  P. 

5.  Find  the  envelope  of  the  lines  y  —  /3  =  7n(x  —  a)  +  rVl  +  m^,  m 
being  a  variable  parameter. 

6.  What  is  the  evolute  of  the  envelope  of  Ex.  5? 

7.  Circles  are  described  on  successive  double  ordinates  of  a  parabola 
as  diameters;  show  that  their  envelope  is  an  equal  parabola.  Find  what 
part  of  this  system  of  circles  does  not  admit  of  an  envelope. 

8.  Show  that  the  envelope  of 

/(x,  y)a^+<f>{x,  y)a  +  \l/(x,  y)  =0 

9.  Find  the  curve  whose  tangents  have  the  general  equation 


y  =  mx  ±  -s/arn^  +  bm  +  c. 

10.  Prove  that  the  circles  which  pass  through  the  origin  and  have 
their  centers  on  the  equilateral  h  v'perbola 

x^  ~  y^  =  a^ 
envelop  the  lemniscata  (x^  +  y^)^  =  4  a\x^  —  y"^. 

11.  If  in  Ex.  10  the  locus  of  the  centers  of  circles  passing  through 
the  origin  be  the  parabola  y^  =  4az,  the  envelope  will  be  the  cissoid 

y\x  +  2  a)  -X*  =  0. 

12.  Show  that  a  family  of  curves  having  two  independent  parameters 
has  no  envelope. 

13.  In  the  "  nodal  family"  {y  -2  xy  -  {x  -  xY -\-%x*  -  y*,  show 
that  the  usual  process  gives  for  envelope  a  composite  locus,  made  up  of 
the  "node-locus  "  (a  line)  and  envelope  proper  (an  ellipse).     Generalize. 


APPENDIX 


oXKo 


NOTE  A   (P.  29) 

Let  y=f(x)  be  a  function  which  is  continuous  and  increasing 
from  x  =  ato  x=  b;  and  let  /(a)  =  A,  f(h)  =  B. 

Let  the  inverse  function  be  written  x=  ^{y);  then  it  is  pro- 
posed to  sliow  that  ^{y)  is  a  continuous  function  of  y  from 
y  =  A  to  y  =  B. 

Let  h  be  any  assigned  positive  number  numerically  less  than 
b  —  a;  then,  since  f(x)  is  an  increasing  function, 

f(x  +  h)-f(x) 

preserves  its  sign  unchanged  when  x  and  x  +  h  both  lie  anywhere 
in  the  interval  from  a  to  b.  Let  the  smallest  value  that  this  dif- 
ference can  take  for  the  assigned  value  of  ^  be 

f(x-^h)-f{x)  =  ky  (1) 

then  when  a;'  >  a;  +  ^, 

f(x')-f{x)>k.  (2) 

Consequently,  if 

f(x')-fix)<k,  (3) 

then  x'  must  be  less  than  x  -{-  h, 

i.e.,  x'  —  x<.h;  (4) 

or,  putting  f(x)  =  y,  f(x')  =  y',  x  =  <f>  (y),  »'  =  </>  (j/*), 

(3)  and  (4)  may  be  written  thus : 

if  y'-y<k,  (5) 

then  <i>  {y')  —  <f>(y)  <  h,  the  assigned  number.  (6) 

314 


Notes  A-B.]  APPENDIX  315 

Hence,  <^  {y)  is  a  continuous  function  of  y  throughout  the  stated 
interval.  A  similar  proof  applies  to  intervals  in  which  /(«)  is  a 
decreasing  function.     Hence : 

For  every  interval  in  ivhich  a  function  is  continuous  there  exists 
an  intei'vcd  in  which  the  inverse  function  is  continuous. 

NOTE   B  (P.  60) 

To  prove  ^™gc(  ^  "I — J  ~  ^'  tvhen  m  is  a  positive  integer. 

The  proof  of  Art.  30  can  be  readily  completed  by  use  of  a 
method  exemplified  later  in  Art.  67.  As  shown  in  Art.  30  the 
problem  is  to  prove  rigorously  that  the  limit,  when  m  =  oo,  of  the 
sum  of  the  entire  m  + 1  terms  of  the  series 

is  equal  to  the  sum  to  infinity  of  the  series 

^+i"^r:2"^r:2T3"^i.2.3.4'^"*'  ^^^ 

without  unduly  applying  the  theorems  of  limits  in  the  case  of  an 
infinite  number  of  variables.  For  this  purpose  the  remainder  of 
series  (1)  after  the  first  n  terms  will  now  be  examined. 

Let  the  sum  of  the  first  n  terms  in  (1)  and  (2)  be  denoted  by 
S^  and  E^  respectively,  then 

1-1  1-—     1     ^-^ 

„       ^      ^      1         wi  .         .  1         m  m  XQ. 

^,  =  l  +  l  +  j-2-+-+i^ ^31-'  (3) 

jE;„  =  H-1+  — H h ^ ;  (4) 

1.2  1.2...n-l*  ^^ 

and,  evidently,  when  n  is  any  finite  number, 

JL"^^S„  =  ^,.  (6) 

Next  let  7?„  be  the  remainder  of  the  series  (1)  after  its  first  n 

terms,  that  is,  the  sum  of  the  last  m  +  \  —  n  terms ;  then  the 

sum  of  the  series  is 

S^S^+R^  .  (6) 


316  DIFFERENTIAL   CALCULUS  [Note  B. 

and  mT^^-m  Too  ^'^  =  m  ™oo  ^»-  C^) 

Now  the  first  term  in  R„  is  the  (n  +  l)st  term  of  series  (1),  and 
the  ratio  of  this  term  to  the  preceding  (which  is  the  last  term 
in  (3))  is 

1  _  ^  ~  1 
m 


n 

but  this  ratio  is  less  than  -,  and  evidently  the  ratio  of  any  subse- 

n 

quent  term  to  the  preceding  one  is  still  less  than  this,  therefore 

Rn<uJ'^  +  \  +  \+-\  [Cf.p.ll2. 

\n     w     w         ) 


l^ence  /f^  ^"  <  ^»(  r^)  <  .„      ..!„      -.^.-  (») 


lim 

,n-iy      (n-l)(n-l)I 

It  follows  from  (5),  (7),  (8)  that 

and  therefore  that  this  difference  can  be  made  as  small  as  desired 
by  taking  n  large  enough.  Thus  the  limit  when  m  =  go  of  the 
sum  of  series  (1)  is  equal  to  the  limit  approached  by  the  sum  of 
the  first  n  terms  of  series  (2)  when  n  is  infinitely  increased;  and 
this  completes  the  proof  of  Art.  30,  when  m  is  a  positive  integer. 

To  prove  the.  theorem  when  m  is  unrestricted. 
If  m  is  positive  but  not  an  integer,  let  it  be  supposed  to  lie 
between  the  two  positive  integers  p  and p  + 1,  i.e. p<m<p  +  l; 


(1) 


then 
hence 

1, 

P' 

m                 p            m 

(i+lY.A+iY- 
\    pJ  \    my 

(-S'>(- 

my 

Again, 

1      1     A     1  V*'   /-.    ly*' 

) 

hence 

A+  1  Y^'.A  +  i^ 

r-'<[i+Lr. 

(2) 


Note  B.]  APPENDIX  317 

Hence,  from  (1)  and  (2), 

It  will  now  be  shown  that  when  jh  m,  p  -\-l  all  =  co,  the  first 
and  third  members  of  these  inequalities  have  the  common  limit  e. 
For,  since  the  exponents  m  —p,  m—p  —  l  are  finite, 

\       mj  \       mj 

but  since  p,  p  +  i  are  infinite  positive  integers, 

\p+i 


(-^)'-'  d^j^J 


=  e\ 


hence  e  is  the  common  limit  of  the  first  and  last  members  of  (3), 
and  is  therefore  also  the  limit  of  the  intermediate  member, 


ie  "°^ 


=  "1,       mj 
Finally,  let  m  be  any  negative  number,  say  —p, 

Writing  k  for  p  —  1, 
but  when  m  =  —  co  and  A:  =  +  oo. 


=  e 
therefore,  by  (4), 


lim 
m 


318  DIFFERENTIAL   CALCULUS  [Note  C. 

NOTE  C   (P.  187) 

On  maxima  and  minima  in  two  variables. 

In  giving  the  criteria  for  maxima  and  minima  in  Art.  107  it 
was  stated  that  it  is  in  general  unnecessary  to  consider  terms 
above  the  second  degree  in  h  and  k,  as  such  terms  are  usually 
infinitesimals  of  an  order  higher  than  that  of  the  second  degree 
terms.  The  exceptional  cases,  in  which  some  of  the  terms  of 
higher  degree  may  become  of  equal  importance  with  the  second 
degree  terms,  can  be  readily  treated  by  the  method  of  comparison 
illustrated  so  extensively  in  the  later  chapter  on  curve  tracing. 

Using  the  notation  of  Art.  107,  let 

^(f>  =  <f>{a  +  h,  b  +  k)  —  <f>{a,  b)  =  U2  +  u^  +  tt^  H ,        (1) 

in  which  u^  denotes  a  homogeneous  polynomial  in  h  and  k  of 
degree  r;  and,  representing  the  function  <l>(x,  y)  as  usual  by  the 
ordinate  of  the  surface  whose  equation  is  z  =  <^  (x,  y),  let  the 
origin  be  transferred  to  the  critical  point  whose  coordinates 
are  a,  b,  ^  (a,  b) ;  then  the  equation  of  the  surface  becomes 

Z'  =  A<i>  =  U2  +  Us  +  Ui-\ ,  (2) 

in  which  h,  k,  z'  are  the  new  current  coordinates.  The  equation 
of  the  tangent  plane  at  the  origin  is  then  z'  =  0,  and  the  curve  of 
section  which  it  makes  on  the  surface  has  the  equation 

Wj  +  M8  +  W4+-=0.  (3) 

The  form  of  this  plane  curve  in  the  vicinity  of  the  origin  will 
be  a  decisive  test  for  a  maximum  or  minimum.  By  Chapters 
XVII,  XVIII,  when  the  lowest  terms  are  of  the  second  degree, 
the  origin  is  either  a  node,  a  cusp,  a  point  of  osculation,  or  a  con- 
jugate point.  If  the  factors  of  Mg  ^^'^  imaginary,  the  origin  is 
an  isolated  or  conjugate  point  of  the  locus,  hence,  in  the  vicinity 
of  the  critical  point,  the  surface  is  altogether  at  one  side  of  the 
tangent  plane,  and  has  a  maximum  or  minimum  ordinate.  If  the 
factors  of  Wg  ^'^e  real  and  distinct,  the  curve  of  section  has  two 


Note  C]  APPENDIX  319 

branches  passing  through  the  origin,  hence  part  of  the  surface 
will  be  above  the  tangent  plane  and  part  below  it,  and  there  will 
thus  be  no  complete  maximum  or  minimum.  In  both  of  these 
cases  it  is  unnecessary  to  examine  the  higher  terms  unless  a 
minute  knowledge  of  the  deportment  of  the  given  function  is 
desired. 

Lastly,  let  U2  be  a  complete  square  of  the  form  (Ah  +  Bky. 
In  this  case,  the  origin  is  usually  either  a  cusp  or  a  point  of  oscu- 
lation, as  in  Art.  163 ;  but  it  may  possibly  be  a  conjugate  point 
of  the  kind  noticed  in  Ex.  9,  Art.  164,  at  which  the  tangents  are 
real  and  coincident.  It  is  therefore  necessary  to  examine  the 
higher  terms.  For  convenience  transform  the  axes  so  that 
h' =  Ah -{- Bk,  k' =  Bh  —  Ak,  then  the  equation  of  the  curve 
takes  the  simple  form 

^"+w'3  +  «',+  ...=0.  (4) 

When  the  method  of  comparison  of  Art.  170  is  applied,  suppose 
it  is  found  that  h'  and  A;'*"  are  of  the  same  order,  then  all  the  terms 
of  (4)  that  are  of  the  same  order  as  h"^  will  constitute  a  poly- 
nomial in  h'  and  k'",  which  can,  as  in  Art.  170,  be  factored  into 
the  form  (h' -^  fik''')(h' +  vk'"').  These  will  be  the  controlling 
terms ;  hence,  when  /m,  v  are  imaginary,  the  origin  is  a  conjugate 
point,  and  there  is  a  maximum  or  minimum  ordinate  of  the  sur- 
face ;  but  when  /x,  v  are  real  and  distinct,  the  origin  is  a  cusp  or  a 
point  of  osculation,  according  as  m  is  or  is  not  a  fraction  with 
even  denominator,  and  there  is  no  complete  maximum  or  mini- 
mum. When  fi,  V  are  real  and  equal,  the  above  process  is  to  be 
repeated.     For  a  simple  illustration  see  Ex.  5,  p.  190. 

Ex.  1.   Show  that  when  c  ■<  1,  unity  is  a  turning  value  of 

«  =  H-(x  -1-  y)«  +(a;  -t-  y)(x'^  +  a;y  +  y2)+  4cjc«. 

Ex.  2.  For  different  values  of  c,  examine  in  the  vicinity  of  the  values 
a;  =  0,  2/  =  0,  the  deportment  of  the  function 

C  (««  -I-  3  y3)2  _  4  (a;2  +  3  y8)  (jc8  4.  5  y4)  4.  2  (x3  +  5  y4)a. 


320  DIFFERENTIAL   CALCULUS 

NOTE  ON   HYPERBOLIC   FUNCTIONS 

Definitions  and  direct  inferences.  For  the  present  purpose  the 
hyperbolic  cosine  and  sine  may  be  defined  analytically  in  terms 
of  the  exponential  function,  as  follows : 

cosh  X  =  |(e^  +  e  ')>  ^"^^^  ^  =  \if  ~  ^~')i  0-) 

and  the  hyperbolic  tangent,  cotangent,  secant,  and  cosecant  are 
then  defined  by  the  equations 


,     ,          sinhcc  ,,          cosh  a; 

tanh  X  =  — - — ,  coth  x  =  -:— - — , 

cosh  X  sinh  x 

sech  X  =  — : — ,  csch  x  =  ■ 


cosh  X  sinh  x 


(2) 


Among  the  six  functions  there  are  five  independent  relations, 
so  that  when  the  numerical  value  of  one  of  the  functions  is  given, 
the  values  of  the  other  five  can  be  found.  Four  of  these  relations 
consist  of  the  four  defining  equations  (2).  The  fifth  is  derived 
from  (1)  by  squaring  and  subtracting,  giving 

cosh^  X  —  sinh^  x  =  l.  (3) 

By  a  combination  of  some  of  these  equations  other  subsidiary 
relations  may  be  obtained ;  thus,  on  dividing  (3)  successively  by 
cosh^ic,  sinh^a;,  and  applying  (2),  it  follows  that 

1  —  tanh'''  X  =  sech^a;. 


(4) 

coth^  a;  —  1  =  csch^  x. 

Equations  (2),  (3),  (4)  will  readily  serve  to  express  the  value 
of  any  function  in  terms  of  any  other.  For  example,  when 
tanh  a;  is  given, 

coth  X  = ; — ,    sech  x  =  VI  —  tanh^  x, 

tanh  X 

coshx  = J- .    sinh  a.  =        *^"^«^ 


Vl  —  tanh^a;  Vl  —  tanh^a; 

Ex.  1.   From  equations  (1)  prove 

cosh  (  —  at)  =  cosh  x,  sinh  (  —  aj)  =  —  sinh  x, 

coshO=  1,  sinhO  =  0,  coshoo  =  oo,  sinhoo  =  oo. 


APPENDIX  321 

Ex.  2.   From  equations  (3),  (4)  show  that 

cosh  a;  >  siuh  x,  cosh  a;  >  1 ,  tanh  a;  <  1 . 

Ex.  3.   Prove  that  tanh  «  =  ±  1,  when  a;  =  ±  oo. 

Ex.  4.    By  direct  substitution  from  (1)  verify  the  addition  formulas 

sinh  {X  ±y)  =  sinh  x  cosh  y  ±  cosh  x  sinh  y, 

cosh  (x  ±  y)  =  cosh  X  cosh  y  ±  sinh  x  sinh  y  ; 

and  hence  derive  the  conversion  formulas 

cosh  X  +  cosh  y  =  2  cosh  \(x  +  y)  cosh  K*  —  ?/)> 

sinhx  +  sinh  y  =  2  sinh  i(*  +  J/)  cosh  \{x  —  y);  etc. 

Show  that  the  corresponding  formulas  for  the  circular  functions  could  be 
verified  by  their  exponential  expressions  (p.  101). 

Ex.  6.    Prove  the  identities  :  sinh  2  x  =  2  sinh  x  cosh  x, 

cosh  2  X  =  cosh2  x  +  sinh^  x  =  1  +  2  sinh2  x  =  2  cosh^  x  —  1. 

Ex.  6.    Prove  cosh  nx,  +  sinh  nx  =  e"*  =  (cosh  x  +  sinh  x)". 

Derivatives  of  hyperbolic  functions.     By  differentiating  (1), 

—  cosh  x=\{e'  —  e~')  —  sinh x,  (5) 

-^  sinh  X  =  1^  (e'  -f-  e"')  =  cosh  x ;  (6) 

dx 

hence      A  tanh  x  = -^  glH^  =  ^"^^' ^  -  f  ^^^  ^  ^  sech'' or.         (7) 
die  dx  cosh  a;  cosh-  a; 

d?  d^ 

Also,  —  cosh  X  =  cosh  «,  — -  sinh  x  =  sinh  x.  (8) 

'  d^  da;2  V  / 

d?  d^    •  • 

—  cosh  mx  =  m^  cosh  mx,  — —  sinh  mx  =  m^  sinh  mx.       (9) 
dx2  '  dx-2  ^  ^ 

It  thus  appears  that  the  functions  sinh  x,  cosh  x  reproduce 
themselves  in  two  differentiations,  just  as  the  functions  sin  x, 
cos  a;  produce  their  opposites  in  two  differentiations.  In  this 
connection  it  may  be  noted  that  the  frequent  appearance  of  the 
hyperbolic  (and  circular)  functions  in  the  solution  of  physical 
problems  is  due  to  the  fact  that  they  answer  the  question :  What 
function  has  its  second  derivative  equal  to  a  positive  (or  negative) 
constant  multiple  of  the  function  itself  ? 


322  DIFFERENTIAL   CALCULUS 

Ex.  7.    Eliminate  the  constants   by  differentiation   from   the  equation 
y  =  A  cosh  mx  +  B  sinh  mx,  and  prove  — ^  =  iiV-y. 

Ex.  8.    Prove  —  coth  x  =  —  csch^  a;,   ^  sech  x  —  sech  x  tanh  a;, 
dx  dx 

Expansions.     By  applying  Maclaurin's  theorem,  using  (5),  (6), 

(8),  or  else  by  substituting  the  developments  of  e*,  e'"',  in  (1),  the 

following  series  are  obtained: 


/v2  /v»4  /«D 

coshx=l+-  +  -  +  gj  + 

3*^  if  iK 

sinha.  =  a-+3y  +  ^  +  -  + 


(10) 


By  means  of  these  series,  which  are  available  for  all  finite 
values  of  x,  the  numerical  values  of  cosh  x,  sinh  x  can  be  com- 
puted and  tabulated  for  successive  values  of  x* 

Derivatives  of  inverse  hyperholic  functions. 

Let  y  =  sinh~^  x,  then  x  =  sinh  y ; 


clx  =  cosh  ydy  =  Vl  +  x^dy; 

hence  —  sinh~^  x  =  —  •  (11) 

dx  Vl  +  ar' 

Similarly,  — cosh"^a;  =  —  •  (12) 

^  dx  Va:^-1 

Again,  let  y  =  tanh~^  a;,  then  x  =  tanh  y, 

da;  =  sech^  y  dy  =  (1  —  tanh^  y)  dy  =(1  —  a?)  dy ; 


therefore  —  tanh~^  x  = 

dx  1  — arj, 


(13) 


Similarly, 


—  coth-^  X  =  -T^l      •  (14) 

dx  x^-  lj^>i 


Ex.  9.    Prove       —  sech-i  x  =  —  cosh-i  1  -  ^ 


dx  dx  x     ccVl  —  x2 

—  csch"^  X  =  —  sinh-i  -  =  — ~ 

dx  dx  X     jcVl  +  x'^ 


*  See  Tables,  p.  162,  Memman  and  Woodward's  "  Higher  Mathematics." 


APPENDIX  323 


Ex.  10.   Prove*     dsinh-i^=       ^      ,  dcosh-i^  =  -      <^« 


adx   ^         ,7„„ii,-ia;  «(?x 

Jx<a 


Relation  of  hyperbolic  functions  to  hyperbolic  sectors. 

In  the  circle  a^  -\-y^  =  a%  let  ^  be  the  area  of  the  sector  in- 
cluded between  the  radii  drawn  to  the  points  (a,  0),  (x,  y);  and 
let  6  be  the  included  angle ;  then,  by  geometry, 

2  A=a^6=a^  sin~^-^  =  a^  cos  ~'  — 
a  a 

Again,  it  is  shown  in  Integral  Calculus  by  means  of  the  deriva- 
tives in  Ex.  10,  that  in  the  hyperbola  a^  —  y^=  a^,  if  A'  be  the 
area  of  the  sector  included  between  the  radii  drawn  to  the  points 

(a,  0),  (x, y),  then  2  A'=  a^  sinh~^^  =  a^  cosh"^-. 

a  a 

Thus  the  hyperbolic  functions  are  related  to  hyperbolic  sectors 

as  the  circular  functions  are  related  to  circular  (and  elliptic) 

sectors,  t 

Expansions  of  inverse  hyperbolic  functions. 
By  the  method  of  Art.  67, 

8inh-ia;=a:-^  |  +  1 1|_....  [-l<a.<l]         (15) 

Another  series,  convergent  when  a;  >  1,  is  obtained  by  writing  the 
derivative  in  the  form 

-^  sinh-'a;  =  (a^  +  1)  "^  =  1  (l+K"'^ 
dx  x\       of . 


x\         2ar'     2  4a^         / 


hence,  8inh-'a.=^+logaj+|  ^^-1 1  A. +...,  (16) 


*  These  derivatives  will  be  found  useful  in  the  "Integral  Calculus." 
t  For  a  treatment  of  the  hyperbolic  functions  from  this  point  of  view, 
see  Merriman  and  Woodward. 

DIFF.  CALC. 22 


324 


DIFFERENTIAL   CALCULUS 


where  -4  is  a  constant,  which  is  shown  later  to  be  equal  to  log  2 ; 

11        13    1 


similarly,      cosh-'«=^+loga;  — 


2  2  af^      2  4  4  o;^ 


(17) 


which  is  always  available  for  computation,  since  cosh~'a;  is  a  real 
number  only  when  x  >  1. 

Ex.  11.     Prove  that  tanh-^x  =  x+«3+-x^+  •••,  and  that  this  series  is 

3         5 

always  available  when  tanh-'x  is  real,  i.e.,  when  —1  <  x<  1. 
Logarithmic  expressions  for  inverse  hyperbolic  functions. 


Let 

hence 

Similarly, 

Also 

Again,  let 

therefore 

i.e., 

hence 

and 


;=cosh?/,  then  Vaj^— l=sinh?/. 


05+ Var^— 1  =  cosh  ?/+ sinh  2/=^, 
2/=cosh"'a;,=log(x+ Va^— 1). 


sinh"^a;=log(a;+  Va^+1). 
sech"^ic=cosh 


X  X 


csch"^a;=sinh-^-  =  log -^t ±_. 


a;=tanhy= 

1+a;^  e^ 
l—x~e-' 


=  e^ 


2y=log-^i-; 
1— X 

tanh~^a;=  -  log  -i-, 
2        l~x 

coth-ia;=tanh-^-  =  \  log  ^±^- 


X     2 


x-l 


Ex.  12.    Show  from  (18),  (19)  that,  when  x  =oo , 

sinh-i «  —  log  X  =  log  2,      cosh-i  x  —  log  x  =  log  2, 
and  hence  that  the  constant  vl  in  (16),  (17)  is  equal  to  log  2. 


(18) 
(19) 
(20) 

(21) 


(22) 
(23) 


APPENDIX  325 

Graphs  of  hyperbolic  functions.  The  student  is  advised  to 
sketch  the  graphs  of  these  functions  from  their  definitions  and 
fundamental  properties.  Aid  is  also  obtained  from  the  values 
of  their  first  and  second  derivatives. 

Ex.  13.  The  curve  y  =  siiih  x  has  an  inflexion  at  the  origin,  the  slope  of 
the  tangent  being  unity  ;  the  bending  is  upwards  to  the  right  and  downwards 
to  the  left  of  the  origin. 

Ex.  14.  The  curve  y  =  cosh  x  is  symmetrical  as  to  the  ^-axis,  and  has  a 
minimum  ordinate  at  x  =  0. 

Ex.  15.   Show  that  the  curve  y  =  tanh  x  has  two  asymptotes  y  =  ±1. 

Ex.  16.   Using  the  graphs,  give  approximate  solutions  of  the  transcendental 

3 
equations ,    tanh  x  =  x  —  1 ,    cosh  x  =  x  +  2,     sinh  x  =  -  x,     cos  x  cosh  x = 1 . 

Ex.  17.  The  equation  of  the  catenary  is  -  =  cosh  - ;  show  that  the  deriva- 

tive  of  the  arc  is  ^p-  =  cosh  - ,  and  hence  that  -  =  sinh  -  • 
dx  c  c  c 

Chidermanian  function.  When  two  variables  x,  y  are  so  related 
that  secy  =  cosh  a;,  then  y  is  called  the  Gndermanian  function  of  x, 
and  is  denoted  by  gd  x.  The  angle  whose  radian  measure  is  equal 
to  grd  a;  is  called  the  Gudermanian  angle  of  x. 

Ex.  18.  Show  that  the  six  hyperbolic  functions  of  x  can  be  expressed  as 
circular  functions  of  gdx:  e.g. ,  cosh  x  =  sec  gd  x,  sinh  x  =  tan  gd  x,  etc. 

Ex.  19.    The  curve  y  =  gdx  has  asymptotes  y  =  ±  i  ir. 

Ex.  20.    Prove  —  gdx  =  sech  x,  —  gd-^  x  =  sec  x. 
dx  dx 


NOTE  ON  INTERPOLATION  BY  TAYLOR'S 
THEOREM 

■ 

7\no  ordinates  given  ;  to  compute  an  intermediate  ordinate.  In 
the  ciu-ve  y  =  f(x),  let  the  ordinate  at  x  =  a  be  yi,  and  let  the 
ordinate  at  x  =  a  +  h  be  ^2-  If  Vn  Vi  b^  given  numerically,  it  is 
required  to  compute  the  ordinate  y  at  the  intermediate  point 
x  =  a  +  fh,  where  €  <  1. 


326  t>IfrERt:NTtAL   CALCULUS 

Consider  the  three  equations, 

yx  =/(«),  (1) 

y,=f(a  +  h)=f(a)-thf'{a),  [neglecting /iy"(a)]  (2) 

y=f{a  +  eh)  =f(a)  +  ehf  (a) ;  (3) 
then  from  (1),  (2),  hf'(a)  =  2^2  —  2/i  j  hence,  by  (3), 

y  =  yi  +  ((2/'2-yi)'  (^) 

The  neglect  of  the  term  h^f"{a)  in  (2)  is  justified  either  when 
h^  is  very  small,  or  when  /"(a)  is  zero.  The  former  is  the  case 
when  the  given  ordinates  are  very  close  together.  The  latter  is 
the  case  when  f(x)  is  of  the  first  degree,  i.e.,  when  the  locus 
y  =f(x)  is  a  straight  line;  hence  (A)  gives  accurately  the  ordinate 
of  the  straight  line  joining  two  given  points  on  a  curve. 

TJiree  equidistant  ordinates  given  ;  to  compute  an  intermediate 
ordinate.  Let  the  ordinate  at  a  —  ^  be  y^,  at  a  be  y^,  at  a  -f  ^  be 
y^;  it  is  required  to  find  the  ordinate  at  a  -f-  eh,  where  — l<c<l. 
In  this  case,  neglecting  h^f"{a), 

2/1  =/(«  -  h)  =  f(a)-  hf<{a)+\hr'{a),  (4) 

y,=f{a  +  h)  =  f{a)+  hf{a)+\hr\a),  (6) 

y  =f(a  +  cA)=/(a)+  c^/'(a)+  i£Viy"(a).  (7) 

From         (4),  (5),  (6),    y,-2y,  + y,  =  hr'{ay,  (8) 

and  from       (4),  (6),  •         y,-y,  =2hf'(a)i  (9) 

hence,  substituting  for  hf'(a),  h^f"(a),  in  (7), 

y  =  y2  +  ^^(ys-yi)+^^(yi-2y2  +  ys)-     -        (-B) 

This  interpolation  formula  gives  accurately  the  ordinate  of  a 
parabola  whose  equation  is  of  the  form  y  =  A  +  Bx-\-  Cx^. 
Four  equidistant  ordinates.     The  result,  similarly  found,  is 

y  =  h(yi  +  y*)-^(2q-  r)  +  \e{y^-y^-  ^^^r) 

+  ^e\2q-r)-^^r,         (C) 

in  which       9  =  2/1-2^2  +  2/3,  r  =  2/i-32/x  +  32/3-2/4- 


ANSWERS 


Art.  16 

1.  2x-2. 

2.  6x-4.               3. 
Art.  18 

1 

4x2 

4.  4x3  _1. 

X8 

1.  15y2_2. 

2.  14«-4-33«2. 

Art.  20 

3.  12 1*2 -2. 

4.  4  X  -  6. 

2.  Inc.  from  —  co  to  | ;  dec.  from  |^  to  1 ;  inc.  from  1  to  +  co  ;  |^  and  ;•.. 
8.  Two.     +latx  =  i±\/S;  -latx  =  i± v^. 
4.  ±  tan-i  ^. 

2.  72x5-204x2. 
1.  -^. 
a? 


Art. 

21 

Art. 

28 

6. 

ny 

xVl  - 

x2 

7. 

4-a^ 

+  3; 

x^ 

(a2  -  x2)*  4v^  V«*  +  a;* 

8.  (x-6)(x-c)2  +  (x-ffl)(x-c)2 
3    _  m(ft  +  x^  +  n(a  +  x),  ^  2(x  -  a)(x  -  6)(x  -  c). 


(o  +  x)'"+i  (6  +  x)»+» 


g  2x»-4x 


4.  \/a(A/x-\/a)  (l-x2)*(l+x2)^ 
2y/x  Vx  +  a  ( Va  +  Vx)*                 j^    -  2  nx"-\ 

(X»-l)2' 

5.   \ 13     -  hH  _  bx 


Vl-.x2(l-x)  ■     a2y  aVS231^ 

Art.  33 

1  8x-7  _1_ 

4x2-7x  +  2  3. ^^"^'    . 

(l  +  x)2 

2.  46**+".  4.   x»-i  +  nx»-Uogx. 

327 


328  ANSIVERS 

6.  1  10.   x'e=^(l  +  logos). 
2(v'x  +  l)  11.   2xa^-\oga. 

6-    1-2/'-                       •  j2    ^_  _          12x^2+^-1 

7.    ^1^.  '   l«g«  '  2(3x2  -  V'2+^)  V2i:^ 

^^  '^^'^^  IS.   x»a+logx). 

8.  y-3x^'.  3 

1  14     -(X  -  1)^(7x2 +  30x- 97) 


9 


« logx  12(x  -  2)^(x  -  3)^^ 


Art, 

.  40 

1. 

10  X  cos  5  x''. 

7.    -2csc2x. 

2. 

14  sin  7  X  cos  7  X. 

8.   n  sin»-i  x  sin  (m  +  l)x. 

8. 
4. 

tan*  X  —  1. 
2  cos  2  X. 

9.   tan2x. 

10.   cos2m^. 
dx 

6. 

1                1 

— r  loga-  a»=.sec 

X 

2(J). 

11.   y(^^^^  +  cosx\ogx\ 

12.   cos  (sin  u)  cos  w  — ■ 
6.   secx.  dx 


Art.  47 

-2x 


1.  sin-ix  + 10. 

vT3^  1  +  (x-2  _  5)2 

2.  sec2xtan-ix  +  -^^2^.  n  1 

3.  ' 


!+*'•  ^/^^^2 


Vl  -  2  X  -  x2  12.    ^A. 

4       2(l-x2)   • 
'    1  +  6  x2  4-  X*  .  o    cos  log  X 


2xV9x-l 
13. 


5. 


X 


1  +^^  14.   cotx. 


6. —  ,.     xcosx'* 


7. 


xvi-(iogx)«  '"•  ;^E^' 

-  1 


CO.S-lx  •   Vl  —  X2  g       '„:„! 

Jo.    sm — 

g  4x  x2         X 


Vl  -4x* 


17. 


9.  '  '+'' 

V2  ax  —  x2  18.    —  cos  (cos  x)  sin  x. 


ANSWERS  329 

Page  72.     Miscellaneous  Exercises 

-2 
8. 


e'  +  e-*  Vl  -  x^ 

4cos(21oga;2-7) 


»©"'(•  ^'°<t)' 


9. 


3.  -2csc2x.  10.    1. 

4.  (2  a;  -  5)  e2x  +  4  (x  +  1)  e»  +  1.  n.   2  tan «  +  e«««'  sec t  taut. 
-     e^(l  —  x)—  1  12.   For  all  values. 

{e'  —  1)"  jg    j.^  y  are  determined  from 

6.-2  xe.-'^  cos  X  —  e-*^  sin  x.  a'^2/=  ±  '>^  and  equation  of  curve. 

7    3.»in-i..>Jsm-i2x  ^      21ogx    \  dy  _     2x.v3  -  Sx^y"  +  12x 

\       »            Vl-4xV  •   dx            3xV-22/xS-5 

Art.  51 

1.  12(x2  _  2  X  +  1).  7.   6  tan*x. 

2.  4[(x-2)e2'  +  (x  +  2)e»].  ^q          cosx 

3.  0.  "    (l-sina;)2 

c  11.        ^"* 


4.   -•  (a2  +  x-^)2 

3a2 


8(e»  -  e-«)  13. 


*•■      (e^  +  e-»)8  ■  4Vx(x-a)^ 


1  ,'^x     10  (B-l)! 

6.    xlogx+  — -— •  14.    "<'    .    ^ 

ox"  X 

-1    ,  (-1V 

L  _  «)» "^  (1+  X) 
4!  „«        -24x 


16. 


(l-a;)6  (l  +  2  2/)6 


17.  ^^  =  4 K2 2/^-1).  31.  ^^T/in-r"^- 

"•   (2-y)^»'  •  ^    y      {ey  +  xy 

,^     -2(5  +  8y2  +  3y^)  g,      -2a^a^y 

"•    ^  '*'•    (y2_ax)8 

24.    (-l)n2n-inl{^^^_\^„^,-^^^^\^,,,}. 

(-l)n-i(n-l)I/        1  J 1 

^-  2  l(a;  +  a)"     (x-a)"/ 


830  ANSWERS 

86.   e'(x  +  n-).  n? 

27.   a"  2eax(«2a;2^2anx  +  n(n-l)).        '    (x  +  l)n+i' 

2(-l)»-i(n-3)l 
*»•    ^^::2 30.    -Se'cosx. 

Art.  57 

2!  6.   -8+4(y-3)+3(y-3)2, 

Art.  64 

Voy      3!.  0-3       U      2    / 
•AJrt.  69  .Art.  70 

1 


a(\/3+  v^) 


1.   1. 


Art.  72 
8-   *•      .  4-   4.  6.    f.  7.  i.  8.  -4. 

Art.  75 


1.    1.  11.    -1. 


2.    2. 


2'  on      2 


5-    tV 

6.  2. 

7.  -.  18- 
n 

8.  -i. 


12.   -.  21.  -3. 
o 

2  22.  m. 

n^'  23.  0. 


30.    -. 


3-   4-  2  22.    ,«.  31.  \. 

*•  -i-  ^®'  :;;;i"  o,  n  32.  0. 


14        V2  24.    30.  83.    ^. 

V3+l'  25.    5.  34-    h 


16.    2.  26.   ^.  35-   i- 

9.   log?.  "•   1-  ''  ''•   '•  !!'   Ta 


&  19.-1  28.    0.  38. 


10.   2.  2V2  29.   0. 


39.   V^. 


1.   e-i. 


Art    77 

»•   ^-  6.   1.  7.    1. 

4.  1.  *  «•   aia,-..o„ 


2-   «  '^-  *  i*  6.  1. 


9.  0,  discont 


ANSWERS 


331 


Art.  86 


1. 

7. 

10. 
12. 

13. 
15. 


6. 

7. 

8. 
12. 

15. 
16. 
18. 


—  5,  min.  ;  —  7,  max.    2.   2,  min.  ;  f,  max.    3.    1,  —  J,  min.  ;  |,  max. 


-,  min. 
e 


8.   e,  max. 


9.   ",  min. 
4 


(n  +  J)t,  max.;  («  —  ^)7r,  min. ;  w  any  integer. 

2  jiw,  min.,  and  also  tan-^  ±  V^  for  angles  in  2  and  3 quarter.    (2  n+  l)w, 
tan~i  ±  \/|,  1  and  4  quarter,  max. 


(2  n  +  1)-,  min.;  sin-i^,  max. 


14.    No  max.  nor  min. 


Min.  foi  value  of  x  wliich  satisfies  the  equation   {x  -  l)e*«- 2x2  =  0. 
It  is  between  1^  and  1^. 


2V  Tfl^^  =  I  vol.  of  cone. 
3\T 


Art.  87 

9.    Isosceles. 

10.  Isosceles. 

11.  Radius  of  circle  is  equal  to  height 


Half  that  of  paraboloid  =  -'^. 

Breadth  =  -^,  thickness  =  ^^ 
V3  V6 

Height  Ls  equal  to  diameter  of  base. 

V6(c+6). 

Sine  of  semi-vertical  angle  is  J. 

One  mile  from  destination. 

Side  parallel  to  wall  is  double  the  other. 


of  rectangle. 


17.    (a^  +  6t)t- 
19.    ■\/2. 
23.  30°. 

27.   ^. 


20. 


3a 


13.    1  and  5. 


Art.  90 

.00145.  8.   24 Vs.  9.    2nh. 

11.   5ir.  12.    2. 

Art.  96 

^g  =  ^/(^'  y)  .  ^  +  M^lll .  ^'(a;)da; ;  substitute  0(x)  for  y. 


10.    ±2. 


3a; 
(fe  _  -  2 
etc       y 

dy  _     ax  +  hy  +  g^ 
dx         hx  +  by  +/ 


52/ 


Art.  97 


4.   ^  =  ^. 
dx     y' 


332  ANSWERS 


1.  ^  =  «+e-h-«'cos 
dx     a 


Art.  99 

olog^N/l- 


?)(^^] 


Art.  105 

1.  x  +  y  +  x2  +  a;y  +  |x8-^a;y2- i|/5.... 

2.  X*  +  4  x=h/  +  6  x^y^  +  ixy^  +  y*. 

3.  Vxtany  +  ^^^^M+A^VisecSy  +  l/' ~  ^'^       tan(y  +  gA:) 

+  ^^sec'(y  +  gA:)  ^^22  V^^^  sec^Cj/  +  OA)  tan  (y  +  SA)  V 
vx+eA  _  / 

4.  -25  +  (x-2)2+(y-3)2  +  (2-l)2, 

Art.  107 

4.   X  =  0,  y  =  0,  '  7.   The  three  parts  are  equal. 

8.   6a^ ;  the  parallelepiped  is  a  cube. 

Q    a;     2/      1  ±  Vl  +  a^  +  62  •       .v         • 

»•   -  =  f  = 5 — T-r^ — ;  with  the  upper  sign  there  is  a  maximum  ; 

a     b  a^  +  b^ 

with  the  lower,  a  minimum. 

10.  X  =  y  =  — ^,  min. ;  x  =  y  =  |,  max. 

2  0 

11.  ^-  12.   ax  =  by  =  cs  =  ll^- 

Art.  108 

„       q262 

^^TP"  Mm.,x  =  ±rt. 

6.   Max. ,  X  =  a v^;  min.,  x  =  0.  7.   Max.  for  x  =  ^. 

2 

Art.  117 

8  8  3\/3      3« 

8.    (a)  xxi  +  yyi  =  c2  ;     (5)  xyi  +  Xiy  =  2  A;2  j 

(c)  (2  xiyi  +  yi2)x  +  (xi2  +  2  xiyi)y  =  3  a' ;  (<?)  y-yi  =  cotXi(x  -  Xi). 
6.   X  =  2  ±  Vf  8.    P  =  v'oxlyl,  o. 

16.   At  (0,  0),  90°.     At  the  other  points,  46°.         16.   2«£iz£f.        17.   t. 

3a  —  X  a 

Art.  120 

8.    Subtangent  =  p  tan  o  ;  subnormal  =  p  cot  a.  4.   90°. 

6.   f  =  0;    <t>  =  2e. 


ANSWERS  333 


Art.  126 


1.  J^p^,  ^V¥^^^  llkvWir^2^  ^(a^-x^). 

^  a^-x^     a  a  a^  ^  ' 

2.  J^iL*,   2VaS,  4,rvV  +  «x,   4,rax.  6.   a,   «!co8^ 

'X  2 

Art.  130 
4.   X  =  0,  y  =  1. 

Art.  131 
6.   y  =  a  ;  X  =  0  ;  and  the  oblique  asymptote  x  +  y  =  a. 

Art.  135 

2.  y  =  0,   ?/  +  X  =  -  1,    y  -x--\. 

3.  x-y  =  -l,   x  +  y=l,  x  +  2y  =  0.  4.   y  =  ». 

Art.  136 

8.   xy  +  a2  =  0,  xhf  -  a^  =  0.  9,  10,  11  are  given  in  text 

Art.  137 

1.  x  =  — a,  y  =  —h,y  =  x-\-b  —  a.  9.  x  +  y  =  |a. 

2.  X  =  —  2  a,  X  =  a.  10.  j/  =  0, 

3.  x  =  ±l,  y=±l.  11.  x  =  0,  y  =  0,  x  +  y  =0.          • 

4.  X  =  y  ±  1,  x  +  y  =±1.  12.  x=±a. 

5.  X  =  ±  a,  X  =  ?/  ±  aV2.  13.  x^  =  0 ;  two  parabolic  branches. 

6.  J/  =  X.              7.  X  =  2  a.  14.  y  =  0. 

8.  X  =  2  a,  X  +  a  =  ±  y.  15.  y  =  0,  x  =  y,  x  =  y  ±  1. 

Art.  139 

1.  Parallel  to  initial  line  ;  a  units  above  it. 

2.  One,  perpendicular  to  initial  line,  at  distance  a  left  of  pole. 

3.  Their  equations  are  p  sin  (  (2  *  +  1)  -^  -  tf  \  =  -  esc  ■[  (2  *  +  1)  -  \  • 

*-•  2a        i      a        I  2  ) 

4.  —  =  ±  cos tf  —  sin ^.  5.   psinO  =  2 a. 
2p 

Art.  143 

8.  X  =(!)«.  a.  6.    (4,  |V3). 


334  ANSWERS 


1.  Second. 

2.  First, 


Art.  156 
10.    ^(^+a)'^  13.    (cMi^'ji 


5.  Second.  s  r^  m  iA  2 
4.   Second.  ^i-            ^j^j Vo^ 

6.  a  =  —  1.  2^2  15.    3  Vaxy. 

7.  Third.  ^^'  "c"  16.   secx. 

Art.  157. 

2             g"  4  — .  6    "(^  -4cosg)^_ 

(m+l)p'»-i*  *  3^  ■       9-6cos» 


2 


Art.  159 

2«^ 
1.   o  =  2  a  +  3  X,   /3  = — . 

va 

9x-2     ^      ./     ,  a\    /x 


3.    a  =  X  +  3(.'M/2)^,    /3  =  y+3(x22/) 


4.  a^x-?' 

c 

5.  (a+i3)*-(a-/3)5=(4a) 


|V2/2-c2,    ^  =  2j/. 


Art.  164 

1.  (0,0),   x±2/=0.  3.    (0,  0),  X  =  0,  2/ =  0. 

2.  (±a,  0),  2(x±ffl)  =  ±V3^,   (0,   -a).  4.    (0,  0),  x  ±  2/ =  0. 

9.    When  it  is  made  numerically  smaller. 

Art.  178 

1.  t/2  =  ^x3.  4.    (x~yy  +  4ky  =  0. 

2.  X*  +  y*  =  ai  5.    (x  -  a)2  +  (y  -  y3)2  =  /«. 

3.  x^  +  y^  =  c^.  6.   The  point  (a,  ^3). 

9.    (4  a2/'2  f-  ftxy  +  cx2)  =  4  ac  —  6*. 


INDEX 


(The  numbers  refer  to  pages) 


Absolute  value,  84. 
Absolutely  convergent,  84. 
Acceleration,  157. 
Actual  velocity,  151. 
Algebraic  expression,  3. 

operation,  1. 
Argument,  4. 
Asymptote,  221. 
Asymptotic  circle,  240. 
Average  curvature,  84. 

velocity,  151. 

Beman,  113. 
Bending,  243. 

Catenary,  211. 
Cauchy,  M. 

Center  of  curvature,  255. 
Change  of  variable,  1118. 
Circle,  asymptotic,  240. 

of  curvature,  255. 
Cissoid,  211. 
Commutative,  2. 

Comparison  of  infinitesimals,  21. 
Computation  of  ir,  111. 
Concave,  downwards,  241. 

upwards,  241 . 
Conditionally  convergent,  86. 
Conjugate  jwint,  281. 
Constant,  7. 
Contact,  2.52. 
Continuity  of  an  algebraic  function,  30. 

of  a',  31. 

of  log  X,  31. 

of  sin  X,  cos  X,  32. 
Continuous  function,  7. 

variable,  7. 
Criteria  for  continuous  function,  29. 
Critical  value,  134. 
Curvature,  27. 
Cusp,  279. 


Decreasing  function,  43. 
Definition  of  continuity,  8,  158. 

of  curvature,  261. 

of  »th  derivative,  73. 
De  Moivre,  101. 
Dependent  variable,  7. 
Derivative,  37. 

of  arc,  216. 

of  area,  40. 

of  surface,  218. 

of  volume,  218. 

partial,  160. 

total,  160. 
Determinate  value,  117. 
Differentiable,  43. 
Differential,  156. 
Differentiation,  41. 

of  inverse  function,  48. 
Discontinuity,  8. 
Distributive,'  3. 

Elementary  forms  of  curves,  289. 
Entire,  4. 
Envelopes,  308. 
Equiangular  spiral,  125. 
Euler,  172. 

theorem,  171. 
Even  contact,  171. 
Evolute,  267. 
Explicit  function,  4. 
Exponential  curve,  211. 

function,  58. 
Expression,  3. 

Family  of  curves,  307. 
Form  of  remainder,  95. 
Function,  4. 

Functional  differentiation,  47. 
Fundamental  problem,  37. 
theorem,  20. 


335 


336 


INDEX 


General  exp.  func,  82. 
Generating  function,  82. 
Geometric  applications,  212. 
illus.  of  a  der.,  39. 

Hyperbolic  branches,  221. 
functions,  318. 

Implicit  function,  4. 
Incommensurable  power,  58. 
Increasing  function,  43. 
Increment,  8. 
Independent  variable,  1. 
Indeterminate  form,  115. 
Infinite,  10. 
Infinitesimal,  10. 
Inflexion,  243. 
Integral  expression,  3. 
Interval  of  convergence,  82,  90. 

of  equivalence,  82. 
Inverse  function,  5,  58. 

operation,  1. 
Involute,  267,  272. 
Irrational,  4. 

Klein,  113. 

Leibnitz,  75. 

theorem,  75. 
Limit,  9. 
Logarithmic  f nnction,  68. 

Maclaurin,  87.  . 

theorem,  ^i --. 

Maximum,  ^32,  185. 

Mean  value,  107. " 

Measure  of  curvature,  261. 
Minimum,  132,  185. 
Modulus,  60. 
Montferier,  113. 
Multiple  point,  277. 

Naperian  base,  60. 

Natural  base,  60. 

Newton,  75. 

Node,  277. 

Non-unique  derivative,  43. 

Normal,  208. 

length,  209. 
Number,  1. 

Oblique  asymptotes,  227. 
Odd  contact,  254. 
Operation,  1. 


Order  of  contact,  252. 

infinite,  lit. 

infinitesimal,  19. 
Osculating  circle,  255. 
Osgood,  82,  85. 

Parabolic  branches,  221,  300. 
Parallel  curves,  272. 
Parameter,  308. 
Partial  derivative,  160. 
Perry,  l.W,  1(;4, 190. 
Polar  coordinates,  212. 

normal  length,  213. 

subnormal,  213. 

subtangent,  213. 

tangent  length,  213. 
Polynomials,  141. 
Principal  infinitesimal,  19. 
Probability  curve,  2;W,  306. 
Process  of  dififereutiation,  42. 

Radius  of  curvature,  255. 

Rate,  152. 

Rational  expression,  4. 

Rectilinear  asymptote,  221. 

Relative  error,  101. 

Remainder,  86. 

Rolle,  85.  ^ 

Semicubical  parabola,  290. 

Series,  81. 

Shanks,  113. 

Simple  exponential  functions,  58. 

Simultaneous  increments,  33. 

Singular  points,  275. 

values,  117,  275. 
Slope  of  a  line,  40. 
Smith,  113. 

Stationary  tangent,  244. 
Stirling,  85. 
Subnormal,  209. 
Subtangent,  209. 
Successive  differentiation,  73. 

operations,  2. 
Sum  of  a  series,  83. 
Surd  expression,  4. 
Symbol  of  approach,  11. 

of  an  increment,  8. 

for  inverse  functions,  5. 

of  a  function,  5. 
Symmetric  expression,  4.  ' 

Table  of  derivatives,  71. 
Tangent,  208. 


INDEX 


337 


Tangent  length,  209. 
Taylor,  87. 

Test  for  convergence,  84. 
Test  for  increasing  function,  45. 
Theorems  on  infinitesimals,  12, 16. 
Total  curvature,  2()1. 
Total  dififerential.  164. 
Tractrix,  211. 

Transcendental,  expression,  3. 
operation,  1. 


Transformed  expression,  4. 
Trigonometric  functions,  58. 
Turning  value,  132. 

Unconditionally  convergent,  89. 
Uniform  velocitj^,  151. 
Unique  derivative,  43. 

Variable,  7. 
Vectorial  angle,  213. 


Typography  by  J.  S.  Gushing  &  Co.,  Norwood,  Mass.,  U.  S.  A. 


QA     McMahon  -  Elements  of  the  differential 
•anji  calculus 

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